This Wikipedia article says that Everson v. Board of Education, 330 U.S. 1 (1947):
...applied the Establishment Clause in the country's Bill of Rights to State law. Prior to this decision, the First Amendment's words, "Congress shall make no law respecting an establishment of religion" imposed limits only on the federal government, ...
However, the Everson opinion does not say that. Rather it quotes Murdock v. Pennsylvania, 319 U.S. 105 (1943) which says:
The First Amendment, which the Fourteenth makes applicable to the states, declares that
"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press . . ."
Everson did say:
Prior to the adoption of the Fourteenth Amendment, the First Amendment did not apply as a restraint against the states. Most of them did soon provide similar constitutional protectionsfor religious liberty.
Murdock goes on to say that:
It is one thing to impose a tax on the income or property of a preacher. It is quite another thing to exact a tax from him for the privilege of delivering a sermon. The tax imposed by the City of Jeannette is a flat license tax, the payment of which is a condition of the exercise of these constitutional privileges. The power to tax the exercise of a privilege is the power to control or suppress its enjoyment. Magnano Co. v. Hamilton, 292 U. S. 40, 292 U. S. 44-45, and cases cited. Those who can tax the exercise of this religious practice can make its exercise so costly as to deprive it of the resources necessary for its maintenance. Those who can tax the privilege of engaging in this form of missionary evangelism can close its doors to all those who do not have a full purse.
and goes on to say
It is contended, however, that the fact that the license tax can suppress or control this activity is unimportant if it does not do so. But that is to disregard the nature of this tax. It is a license tax -- a flat tax imposed on the exercise of a privilege granted by the Bill of Rights. A state may not impose a charge for the enjoyment of a right granted by the Federal Constitution. Thus, it may not exact a license tax for the privilege of carrying on interstate commerce (McGoldrick v. Berwind-White Co., 309 U. S. 33, 309 U. S. 56-58), although it may tax the property used in, or the income derived from, that commerce, so long as those taxes are not discriminatory. Id., p. 309 U. S. 47, and cases cited. A license tax applied to activities guaranteed by the First Amendment would have the same destructive effect.
The Murdock case was not in a school context. It was a case of a license tax on door-to-door sellers of religious literature (Jehovah's Witnesses), which tax the court found unconstitutional under the Establishment clause. It does not proclaim the first incorporation of the Establishment clause, but it does not clearly cite any earlier case that does so incorporate that clause.
Everson v. Board of Education is the earliest case which I can find that applies the Establishment Clause to a state law in the context of public schools. Specifically, it was a case where a NJ law which provided that tax funds be spent to provide transport for children to private schools, including religious schools -- in this case Catholic schools. The Court in Everson took it as unquestionable that the state could not set up religious institutions, including schools, but held that the payment of transportation at issue in the case was permitted, because its primary purpose was to aid the students, not the religious school, it was open to various private schools, not merely religious ones (although in practice only students of Catholic schools benefited in the district sued), and because it aimed to improve public safety, lest children otherwise walk or hitchhike along busy roads.
The court wrote:
The people there [in Virginia], as elsewhere, reached the conviction that individual religious liberty could be achieved best under a government which was stripped of all power to tax, to support, or otherwise to assist any or all religions, or to interfere with the beliefs of any religious individual or group.
The movement toward this end reached its dramatic climax in Virginia in 1785-86 when the Virginia legislative body was about to renew Virginia's tax levy for the support of the established church. Thomas Jefferson and James Madison led the fight against this tax. Madison wrote his great Memorial and Remonstrance against the law. In it, he eloquently argued that a true religion did not need the support of law; that no person, either believer or nonbeliever, should be taxed to support a religious institution of any kind; that the best interest of a society required that the minds of men always be wholly free, and that cruel persecutions were the inevitable result of government-established religions.
Leaving history the Everson opinion goes on to say that:
In recent years, so far as the provision against the establishment of a religion is concerned, the question has most frequently arisen in connection with proposed state aid to church schools and efforts to carry on religious teachings in the public schools in accordance with the tenets of a particular sect. Some churches have either sought or accepted state financial support for their schools. Here again, the efforts to obtain state aid or acceptance of it have not been limited to any one particular faith. The state courts, in the main, have remained faithful to the language of their own constitutional provisions designed to protect religious freedom and to separate religions and governments.
The Everson opinion did not directly consider the case of a government actually running a religious school. It said that the meaning of the Establishment clause from cases applying it to the Federal Government before the passage of the Fourteenth Amendment should also be used in cases applying it to state actions. It cited as earlier cases following that rule:
Cantwell v. Connecticut, 310 U. S. 296; Jamison v. Texas, 318 U. S. 413; Largent v. Texas, 318 U. S. 418; Murdock v. Pennsylvania, supra; West Virginia State Board of Education v. Barnette, 319 U. S. 624; Follett v. McCormick, 321 U. S. 573; Marsh v. Alabama, 326 U. S. 501.
In particular West Virginia State Board of Education v. Barnette was in a school context, specifically about a law requiring school children to recite the Pledge of Allegiance, against claims of some Jehovah's Witnesses that this constituted an act of idolatry, and was religiously forbidden. (Children who refused were expelled, and their parents were prosecuted for failure to have their children attend school)
The Barnette opinion said:
Over a decade ago, Chief Justice Hughes led this Court in holding that the display of a red flag as a symbol of opposition by peaceful and legal means to organized government was protected by the free speech guaranties of the Constitution. Stromberg v. California, 283 U. S. 35
It is now a commonplace that censorship or suppression of expression of opinion is tolerated by our Constitution only when the expression presents a clear and present danger of action of a kind the State is empowered to prevent and punish. It would seem that involuntary affirmation could be commanded only on even more immediate and urgent grounds than silence.
The Fourteenth Amendment, as now applied to the States, protects the citizen against the State itself and all of its creatures -- Boards of Education not excepted.
The very purpose of a Bill of Rights was to withdraw certain subjects from the vicissitudes of political controversy, to place them beyond the reach of majorities and officials, and to establish them as legal principles to be applied by the courts. One's right to life, liberty, and property, to free speech, a free press, freedom of worship and assembly, and other fundamental rights may not be submitted to vote; they depend on the outcome of no elections.
it is important to distinguish between the due process clause of the Fourteenth Amendment as an instrument for transmitting the principles of the First Amendment and those cases in which it is applied for its own sake. The test of legislation which collides with the Fourteenth Amendment, because it also collides with the principles of the First, is much more definite than the test when only the Fourteenth is involved. ... It is important to note that, while it is the Fourteenth Amendment which bears directly upon the State, it is the more specific limiting principles of the First Amendment that finally govern this case.
This is clearly the language of incorporation of the 1st into the 14th against an agency of state government, specifically a school board.
The The Barnette opinion goes on to say:
Probably no deeper division of our people could proceed from any provocation than from finding it necessary to choose what doctrine and whose program public educational officials shall compel youth to unite in embracing. Ultimate futility of such attempts to compel coherence is the lesson of every such effort from the Roman drive to stamp out Christianity as a disturber of its pagan unity, the Inquisition, as a means to religious and dynastic unity, the Siberian exiles as a means to Russian unity, down to the fast failing efforts of our present totalitarian enemies. Those who begin coercive elimination of dissent soon find themselves exterminating dissenters. Compulsory unification of opinion achieves only the unanimity of the graveyard.
If there is any fixed star in our constitutional constellation, it is that no official, high or petty, can prescribe what shall be orthodox in politics, nationalism, religion, or other matters of opinion, or force citizens to confess by word or act their faith therein. If there are any circumstances which permit an exception, they do not now occur to us. [Footnote 19]
We think the action of the local authorities in compelling the flag salute and pledge transcends constitutional limitations on their power, and invades the sphere of intellect and spirit which it is the purpose of the First Amendment to our Constitution to reserve from all official control.
Somewhat later the case of Engel v. Vitale, 370 U.S. 421 (1962) considered an official school prayer used in the public schools of New York state, and held that it was unconstitutional under the Establishment Clause applied to states via the Fourteenth Amendment.
The opinion said:
We think that, by using its public school system to encourage recitation of the Regents' prayer, the State of New York has adopted a practice wholly inconsistent with the Establishment Clause. There can, of course, be no doubt that New York's program of daily classroom invocation of God's blessings as prescribed in the Regents' prayer is a religious activity. It is a solemn avowal of divine faith and supplication for the blessings of the Almighty. The nature of such a prayer has always been religious, none of the respondents has denied this, and the trial court expressly so found[.]
[T]he constitutional prohibition against laws respecting an establishment of religion must at least mean that, in this country, it is no part of the business of government to compose official prayers for any group of the American people to recite as a part of a religious program carried on by government.
That would seem to firmly dispose of any idea of a goverment-run religious school.
As most if not all states had (and still have) their own versions of the Establishment Clause somewhere in their state constitutions prior to the passage of the Federal 14th Amendment, such schools might have been prevented by such state provisions.