Those who argue that the Establishment Clause mandates exclusion of religious institutions from generally available funding programs sometimes argue that such exclusion is needed to protect the religious institutions themselves. Here's an excerpt from Justice Souter's dissent in Zelman v. Simmons-Harris, the 2002 case that upheld the constitutionality of school choice programs that include religious schools:
[I]n the 21st century, the risk [to religion from participation in aid programs] is one of "corrosive secularism" to religious schools, and the specific threat is to the primacy of the schools' mission to educate the children of the faithful according to the unaltered precepts of their faith. Even "[t]he favored religion may be compromised as political figures reshape the religion's beliefs for their own purposes; it may be reformed as government largesse brings government regulation."
The risk is already being realized. In Ohio, for example, a condition of receiving government money under the program is that participating religious schools may not "discriminate on the basis of ... religion," which means the school may not give admission preferences to children who are members of the patron faith; children of a parish are generally consigned to the same admission lotteries as non-believers....
When government aid goes up, so does reliance on it; the only thing likely to go down is independence.... A day will come when religious schools will learn what political leverage can do, just as Ohio's politicians are now getting a lesson in the leverage exercised by religion.
This is a nontrivial policy concern, it seems to me, though on balance I think the value of school choice programs outweighs this risk. But if it's made into a constitutional argument, then I don't see why it wouldn't equally ban the participation of nonreligious speakers in general aid programs. If we take this argument really seriously, then the government should be forbidden from letting students use government-provided grants (Pell grants, GI Bill grants, and the like) in private schools — after all, these grants may come with strings that diminish the "independence" of universities, as historically has indeed happened.
Likewise, the government should be forbidden from letting nonprofit advocacy groups (such as the ACLU, People for the American Way, NARAL, and the like) get tax exemptions, either direct exemptions from the income tax and property taxes, or deductibility for contributions to those groups. After all, such tax exemptions are themselves government subsidies that may come with strings that diminish the "independence" of advocacy groups, as historically has also happened. (For examples, consider antidiscrimination conditions attached both to federal education grants and the tax exemption, which are of course directly parallel to the antidiscrimination conditions attached to school choice funds that Justice Souter complains about; consider also the rule that groups that get tax-exempt contributions may not themselves engage in electioneering or lobbying, but instead have to set up special affiliates for those purposes.)
Of course, these groups are secular groups, and we shouldn't worry as much about their religious freedom. But they are First-Amendment-protected groups, so we should presumably worry about their freedom of speech. Government actions that undermine the independence of religious institutions are bad; but so are government actions that undermine the independence of advocacy groups and universities. (It's true that the Establishment Clause applies only to establishment of religion; but Justice Souter's argument here is focused specifically on harm to the religions' independent practice of religion, which is directly paralleled by the constitutional protection of speakers' independent practice of speech. And on its face the Establishment Clause speaks simply of not establishing religion, or to be precise not enacting laws respecting establishment of religion, not of declining to interfere with the independence of religion. The constitutional constraints on the interference with religious institutions, it seems to me, must closely parallel constitutional constraints on the interference with speaking institutions.)
We should indeed worry about the government using subsidies as a tool for interfering with the independence of universities, advocacy groups, and other speakers. But most liberals (the group most likely to object to the inclusion of religious speakers in generally available funding programs) nonetheless support many such subsidies. Instead of saying that the subsidies are unconstitutional or even on balance unwise, they try to fight undue interference with speakers' independence either (1) through the political process, or (2) through the development of constitutional rules that limit the strings that may be placed on the subsidies. Even libertarians, in my experience, may criticize such subsidies to speakers but don't think that the subsidies are unconstitutional, despite the threat to independence that they may pose.
Yet when the subsidies flow to religious institutions (as part of evenhanded programs that don't discriminate based on religion), many people argue that protecting the independence of religious institutions means that the Constitution requires discriminatory exclusion of those institutions from funding programs. Why would this be so, if protecting the independence of speakers doesn't mean that the Constitution requires exclusion of speakers from funding programs?