The question has a false premise: the law does does not require defendants to pay for counsel. The right to counsel can be waived. However, the fact that court-appointed counsel are provided only to indigent defendants does seem at odds with the text of the Sixth Amendment. I will explain why the right is limited by reviewing the cases and historical background.
Original meaning of the Sixth Amendment
Part of the answer is that the Sixth Amendment was (probably) not originally intended to create a right to court-appointed counsel at all. Rather, it reversed the historical rule that people accused of felonies were denied representation even if they could afford it. See Powell v. Alabama, 287 U.S. 45 (1932), at pp 60–69:
Originally, in England, a person charged with treason or felony was denied the aid of counsel, except in respect of legal questions which the accused himself might suggest ... Historically and in practice, in our own country, at least, [the right to a hearing] has always included the right to the aid of counsel when desired and provided by the party asserting the right.
See also Scott v. Illinois, 440 U.S. 367 (1979), at p 370:
There is considerable doubt that the Sixth Amendment itself, as originally drafted by the Framers of the Bill of Rights, contemplated any guarantee other than the right of an accused in a criminal prosecution in a federal court to employ a lawyer to assist in his defense.
Basis of the modern rule
As noted in the question, Gideon v. Wainwright, 372 U.S. 335 (1963) is the leading case on this area of law today, but the modern rule was more precisely and recently stated in Scott, at pp 373–374:
We therefore hold that the Sixth and Fourteenth Amendments to the United States Constitution require only that no indigent criminal defendant be sentenced to a term of imprisonment unless the State has afforded him the right to assistance of appointed counsel in his defense.
Note that the rule is not derived directly from the Sixth Amendment (at least insofar as it applies to the States). The Bill of Rights did not apply to the States before the Civil War: Barron v. Mayor & City Council of Baltimore, 32 U.S. 243 (1833). After the Reconstruction Amendments, through the process of incorporation, the Supreme Court began to accept that the Fourteenth Amendment's Due Process Clause could entrench substantive rights against the States.
The modern rule recognises that any person who is unable to afford a lawyer and who is sentenced to imprisonment without legal representation has been deprived of their liberty without due process of law. It does not establish an absolute right to legal representation.
In Scott, after noting that the modern rule went beyond what was guaranteed by the text of the Sixth Amendment (understood in its historical context), the Court held that it did not extend to an indigent defendant who was fined for a petty theft. At p 373, the Court offered the following practical justification for limiting the rule:
[W]e believe that the central premise of Argersinger – that actual imprisonment is a penalty different in kind from fines or the mere threat of imprisonment – is eminently sound, and warrants adoption of actual imprisonment as the line defining the constitutional right to appointment of counsel. Argersinger has proved reasonably workable, whereas any extension would create confusion and impose unpredictable, but necessarily substantial, costs on 50 quite diverse States.
An unconstitutional taking?
The Fifth Amendment provides that private property shall not be taken for public use without just compensation. The concept of "taking for public use" is another one that's hard to pin down at the margins, but there's a good summary of the cases available in the section on When Property is Taken in the Constitution Annotated:
The older cases proceeded on the basis that the requirement of just compensation for property taken for public use referred only to "direct appropriation, and not to consequential injuries resulting from the exercise of lawful power." Accordingly, a variety of consequential injuries were held not to constitute takings ... Nor was government held liable for the extra expense which the property owner must obligate in order to ward off the consequence of the governmental action ... But the Court also decided long ago that land can be "taken" in the constitutional sense by physical invasion or occupation by the government, as occurs when the government ﬂoods land permanently or recurrently. A later formulation was that "[p]roperty is taken in the constitutional sense when inroads are made upon an owner's use of it to an extent that, as between private parties, a servitude has been acquired either by agreement or in course of time."
I am not aware of any case in which it was argued that the need to pay for one's own legal representation amounted to a taking for public use. It is certainly a far cry from the compulsory acquisition of land. But perhaps in a case where the defendant is acquitted, or proven innocent, a takings argument could be run by analogy with the cases dealing with innocent people who suffered loss as a result of police execution of search warrants. See Everyone Benefits, Everyone Pays: Does the Fifth Amendment Mandate Compensation When Property is Damaged During the Course of Police Activities? 9 William & Mary Bill of Rights Journal 277 (2000) for more about these cases.