Short Answer
Would a federal tax plan that simply took the amount required to run
the government and divided that amount across the states based on
population and directly charged the states likely to survive a legal
challenge?
A federal tax plan along these lines is expressly authorized by Article I, Section 9, Clause 4 of the United States Constitution, but the federal government has never actually imposed taxes on this basis. But, this has never actually been done because there is no practical way to enforce an obligation of a state to pay a capitation tax.
It has also not been done because it is politically unpopular with members of Congress from states that have a lower per capita income whose support would be necessary to pass such a tax. This is because a tax effectively imposes a higher tax rate relative to ability to pay on poor states than it does on more affluent states and there has always been a significant disparity between more affluent and less affluent states in the United States).
For example, Massachusetts, the U.S. state with the highest per capita GDP ($65,545 in 2018) has more than twice the per capita GDP of Mississippi ($31,881), but would pay the same tax per capita, and, the District of Columbia, which would owe nothing under an Article I, Section 9 capitation tax had a per capita GDP of $160,472 in 2018.
Instead, until the income tax was expressly authorized constitutionally in 1913, the federal government was financed mostly with customs duties and to a lesser extent some select excise taxes (also here and here), with several exceptions, some direct property taxes from 1789 to 1802; a short lived Civil War era income tax on the very affluent (1861-1872), which was reimposed briefly from 1894-1895 before being declared unconstitutional; and some estate taxes, either on the theory that they were excise taxes, or in parallel to the income taxes, until the 16th Amendment's authorization of the federal income tax was found to extend to authorization for federal gift and estate taxation.
As noted here from 1789 to 1860:
Customs duties as set by tariff rates up to 1860 were usually about
80–95% of all federal revenue.
Thus, customs duties were as dominant a source of revenue for the federal government for roughly the first third of the history of the United States, as income and payroll taxes are today.
I have tried to find examples of direct taxation of the states[.]
While what you mean is clear, to avoid confusion, it is important to understand that the phrase "direct taxes" is used in the U.S. Constitution to mean taxes that are imposed directly upon individuals, businesses and private property. The "direct taxation of the states" to which you refer is a form of indirect taxation.
Long Answer
Sources
Most of the discussion in this answer which are not specifically referenced with links or citations to the United States Constitution, is based upon a political science textbook entitled "Congress" that I used in college, an introductory federal income taxation textbook I used as a law school student, and an estate taxation textbook that I used to teach estate taxation when I was a professor for a financial planning program for master's degree students. All are hard copies for which no internet link to the materials themselves is available.
Applicable Constitutional Law
The United States Constitution (which took effect in 1789) has several provisions related to the federal government's power and the power of the states, respectively to impose taxes:
Article I, Section 8, Clause 1 states:
The Congress shall have Power To lay and collect Taxes, Duties,
Imposts and Excises, to pay the Debts and provide for the common
Defence and general Welfare of the United States; but all Duties,
Imposts and Excises shall be uniform throughout the United States;
Imposts and duties are taxes on imports into the United States, while excises are “taxes on the manufacture, sale, or consumption of goods, or upon licenses to pursue certain occupations, or upon corporate privileges[.]"
Among the taxes that aren't "imposts, duties or excises" are income taxes, payroll taxes, gift taxes, estate taxes. For constitutional purposes, payroll taxes, gift taxes, and estate taxes are considered forms of income taxes.
It isn't entirely clear if a general sales tax of the kind adopted by most U.S. states, or a value added tax, is a constitutional excise tax or income tax, as this question has never been tested (such taxes, when collected in the District of Columbia or Puerto Rico or the Virgin Islands by "local governments" fall under the power of the federal government to govern federal territory rather than the taxation power of Article I, Section 8, Clause 1 and the 16th Amendment).
Until 1802 (i.e. for the first 13 years of the current constitution), Congress imposed what amounted to property taxes of various kinds under its excise tax power, but the constitutional validity of these taxes has never been established definitively before they were repealed and those taxes have never been reinstated.
Imposes, duties and excises could be imposed directly by the federal government on private citizens, so long as they were uniform, while "taxes" prior to 1913, were subject to Article I, Section 9, Clause 4 discussed below.
Article I, Section 9, Clause 1 of the U.S. Constitution which was in effect until 1808 when importing slaves was banned stated that a customs duty could be imposed on imported slaves:
The Migration or Importation of such Persons as any of the States now
existing shall think proper to admit, shall not be prohibited by the
Congress prior to the Year one thousand eight hundred and eight, but a
Tax or duty may be imposed on such Importation, not exceeding ten
dollars for each Person.
Article I, Section 9, Clause 4 of the U.S. Constitution states that:
No Capitation, or other direct, Tax shall be laid, unless in
Proportion to the Census or Enumeration herein before directed to be
taken.
The only "tax" other than imposts, duties and excises that could be imposed by the federal government at uniform national rates prior to 1913, was a "capitation" tax imposed directly upon state governments, in proportion to the most recent census.
But, no such such capitation tax was ever imposed upon the states. Congress clearly had and still does have the power to impose such a tax, but it didn't and it still doesn't, for political reasons and for reasons of the enforceability of such a tax. Wikipedia notes that:
The United States government levied direct taxes from time to time
during the 18th and early 19th centuries. It levied direct taxes on
the owners of houses, land, slaves and estates in the late 1790s but
cancelled the taxes in 1802.
And, none of these direct taxes were capitation taxes.
The main reason that this wasn't done is that even though these kinds of taxes were mandatory under the Constitution of the United States adopted in 1789, unlike the 1776 Articles of Confederation which did not have any enforcement mechanism to make states pay their share, enforcement of a capitation tax on states was still impractical in reality.
This is because state governments do not have income or property that is amenable to being seized for payment of taxes in a manner that prevents mass chaos from breaking out (e.g. if the federal government tried to seize a state capitol building or court house for failing to pay its federal capitation tax).
Instead, the premise of a capitation tax is that states will use their taxation power to impose taxes upon their citizens, businesses and the private property in their jurisdiction and then would hand over some of those tax receipts to the federal government until the state's quota for the year, set by Congress, was met.
But, since there is no really viable democratic way for a federal court to direct a state legislature to impose and collect any particular kind of taxes to raise those funds, this nominally mandatory tax obligation of the states was economically and legally impotent, and was still mandatory in name only.
The other problem with a capitation tax is that for almost all of the time period prior to 1913, there was a great disparity between the per capita GDP of the various U.S. states.
For the vast majority of that time period, the commerce and industry of the north (powered to a great extent by hydro power prior to the widespread adoption of coal as a fossil fuel) produced much higher GDP per capita than the agricultural economies of the South, particularly because the infamous "three-fifths compromise" that gave the South Congressional representation based in part upon the number of slaves who lived there (30% or more of the population of many states) did not apply to capitation taxes. After the Civil War, what little commerce, industry and infrastructure that the South had had going in the war was massively degraded, and the population of able bodied men of prime working age was greatly reduced, rendering the South even more poor relative to the North. Even in the heyday of Reconstruction, capitation taxes were not imposed because it was obvious that the South could not afford to pay any significant share of its obligations on that basis.
Thus, the South was mostly poorer per capita to start with, and had a higher share of capitation taxes than it did of Congressional seats and electoral votes, which was not a recipe for building Congressional support for a major new tax.
Because some Southern political support was necessary to impose new taxes for most of U.S. history and because capitation taxes demanded more taxes relative to ability to pay from Southern states than from Northern one (as customs taxes already did to a lesser extent), these taxes never had the political support needed to be adopted by Congress.
Article I, Section 9, Clause 5 of the U.S. Constitution states that:
No Tax or Duty shall be laid on Articles exported from any State.
So custom's duties can be imposed on imports of goods to the United States, but not on exports of goods from the United States. This isn't a big deal today, where only a tiny portion of federal revenues come from customs duties, but it was a big deal prior to the U.S. Civil War.
Historically, this limitation was enacted mostly to prevent the North from burdening the export oriented cash crop agriculture (mostly tobacco and cotton) of the Southern states.
Article I, Section 10, Clause 2 of the U.S. Constitution states that:
No State shall, without the Consent of the Congress, lay any Imposts
or Duties on Imports or Exports, except what may be absolutely
necessary for executing it's inspection Laws: and the net Produce of
all Duties and Imposts, laid by any State on Imports or Exports, shall
be for the Use of the Treasury of the United States; and all such Laws
shall be subject to the Revision and Control of the Congress.
Many people are surprised to learn that states have a right to impose both import and export duties. But, since they can collect no more than the cost of carrying out their inspection laws and must turn the balance of any such revenues over to the federal government, there is no incentive for states to have significant taxes of this type.
The "weigh stations" found on major highways for large trucks, which are mostly used to gather data used to allocate registration fees for trucks among the states based upon economic reality, are the main practical residuary impact of this constitutional provision.
The 16th Amendment to the United States Constitution ratified in 1913 states that:
The Congress shall have power to lay and collect taxes on incomes,
from whatever source derived, without apportionment among the several
States, and without regard to any census or enumeration.
The power of Congress to impose federal income and payroll taxes encompasses employees of state and local governments, as well as private sector employees, even though this means that state and local governments have to file federal withholding tax returns on a regular basis.
But, state and local governments are not taxed on their own income (just like other non-profits) and state and local governments are allowed to establish state and/or local employee retirement systems in lieu of FICA taxation on its employees for employees who participate in those programs.
State and local governments are also not subject to FUTA (federal unemployment taxes). FUTA taxes are in any case de minimis because state unemployment taxes are a credit against federal unemployment taxes dollar for dollar except for a very small minimum per employee FUTA tax (0.6% of the first $7,000 per year of wages for each employee, for a maximum FUTA tax of $42 per year per employee in most cases), because state unemployment taxes are almost always higher than federal FUTA taxes.
Historically, some tax policy makers believed that it was unconstitutional to tax interest from municipal bonds (a.k.a. bonds issues by state and local governments) for federalism reasons. But, case law starting in the late 20th century established that the exclusion of municipal bond interest from taxation is strictly a statutory feature of the Internal Revenue Code and does not have a constitutional dimension. In other words, it is constitutional for the federal government to tax all income derived from municipal bonds, even though it declines to do so for municipal bonds that meet a variety of federal tax law tests.
The History of U.S. Taxation
From 1789 until 1860, the tax revenue of the United States government (as opposed to the states) came predominantly from customs duties on imports and from some select excise taxes (e.g. on alcohol), and in into the mid-1800s with revenues from federal property such as grazing rights and mineral rights, and federal enterprises (mostly the U.S. Postal Service and profits from manufacturing coins and currency).
During the U.S. Civil War, the scope of federal government activities grew dramatically and these were ultimately paid for with an income tax and an estate tax were imposed briefly over constitutional objections but were repealed shortly thereafter, with increased customs duties and excise tax rates, and with confiscation of Confederate property. (The timing was financed with Treasury bonds issued to support the war.)
After the Civil War the scope of U.S. federal government activity returned to pre-war levels, and only started to ramp up again with the Progressive era in the early 1900s followed by World War I, which were financed with the newly authorized federal income tax and an estate tax, at quite low rates by modern standards.
Per the same Wikipedia link referenced above:
Until a United States Supreme Court decision in 1895, all income taxes
were deemed to be excises (i.e., indirect taxes). The Revenue Act of
1861 established the first income tax in the United States, to pay for
the cost of the American Civil War. This income tax was abolished
after the war, in 1872. Another income tax statute in 1894 was
overturned in Pollock v. Farmers' Loan & Trust Co. in 1895, where the
Supreme Court held that income taxes on income from property, such as
rent income, interest income, and dividend income (however excepting
income taxes on income from "occupations and labor" if only for the
reason of not having been challenged in the case, "We have considered
the act only in respect of the tax on income derived from real estate,
and from invested personal property") were to be treated as direct
taxes. Because the statute in question had not apportioned income
taxes on income from property by population, the statute was ruled
unconstitutional.
So, income taxes had been adjudicated by the Courts to be unconstitutional from 1895 to 1913.
Federal income tax withholding taxes are younger than the federal income tax which initially applied with any bite only to very high income individuals (rhetorically, if not mathematically, the equivalent of today's "one percenters"), although the income tax base had widened greatly by the end of the New Deal through World War II and its aftermath.
The U.S. federal government didn't begin to approach its modern scale of activity until the New Deal following the Great Depression in the 1930s, which was financed with very high income taxes and estate taxes, high customs duties such as the Smooth-Hawley tariffs imposed not long after the crash of 1929 (which were so high that they reduced customs revenue rather than increasing it), and newly imposed payroll taxes.
Information tax return reporting (the infamous Form 1099), which dramatically improved income tax collection rates beyond mere wage and salary withholding, was a late 20th century innovation that was adopted based upon the recommendation of free market economist Milton Friedman.
Since the 16th Amendment was adopted, the federal income tax (and later federal payroll taxes on certain kinds of income) became the dominant source of income in short order, with customs duties and excise taxes and other forms of government rents and enterprise income (including income from the Federal Reserve also created before the New Deal in the early 1900s) came to be comparatively insignificant sources of income for the federal government.
One notable exception was that the excise tax on gasoline and other petroleum based fuels, established as a de facto users fee and significant source of revenue to finance the Interstate State Highway system starting in the Eisenhower Administration, did create a significant source of new excise tax revenues. Federally owned oil and gas mineral interests also became a much more important source of federal revenues after World War II.
The automobile had been invented and entered mass production in the early 1900s, but just as the internal combustion engine started to prevail over electric cars at that time and mass production took hold with Ford's Model T bringing some cars and trucks into upper middle class households and into use by businesses and farms, the Great Depression and World War II dramatically reduced private personal consumption, so for the two decades from about 1929 until the several years after World War II that it took to convert factories from war production to civilian peacetime production, the automobile industry's private sector sales were greatly suppressed, domestic civilian road and bridge construction came to a near halt during World War II as government funds were diverted to the war effort, and domestic oil consumption was likewise suppressed.
Footnote On Collecting Debts From State And Local Governments In General
The same issue comes up at the state level when a court awards a money judgment against a state or local government with taxing power, or holds that a state government has insufficiently funded education as required by a state constitution.
Generally, in those cases, a legislative body has to authorize the appropriation "voluntarily" from whatever source of funds it choses to impose, and usually, when push comes to shove, state and local governments do pay the judgments they owe (there are limited federal bankruptcy options for municipalities and states), although there are instances of state governments stubbornly refusing to take the actions necessary to comply with court orders directing that public schools be funded for many, many years.
For the most part, however, because enforcing debts against state governments is so difficult, transactions are structured as much as possible to prevent the need to enforce debts in that way through (1) legal limitations on governmental liability, (2) legislative budget rules requiring interest on debt and currently due principal payments to be made first, (3) third-party bonding of state and local governmental construction projects, (4) the creation of publicly owned corporations whose debts can only be collected out of the corporation's assets and revenues, and (5) avoidance of trade credit obligations by paying bills in cash.
In connection with the legal limitations on liability, it is also worth noting that, as a general rule, state governments cannot be sued for money damages in any federal court other than the U.S. Supreme Court without their permission, and cannot be sued in the U.S. Supreme Court except by the United States government or another state government or perhaps by another country. So, when a money judgment is entered against a state government (which does not include local governments for this purpose) this happens only with multiple layers of state government consent to the process and the exact source and timing of the appropriation to pay it.
When push comes to shove, as they did in California a few years back when it couldn't make payroll, it forced its employees and creditors to accept IOUs in lieu of cash payments, and while they may not have the legal right to do so, they have the economic power to force people to ignore that right.
