You could argue that the 1st Amendment only applies to laws written by Congress and not the whims of state governments but the Supreme Court has ruled in the past that it does extend downwards like that.

Then again, the Supreme Court also ruled that if a state does decide to subsidize private education it can't discriminate based on religion VS non-religion (Espinoza v. Montana Department of Revenue).

The bigger argument: By allowing a state to fund religious institutions (educational or not) you're basically granting the state great power over religious institutions (as well as taking non-sectarian money and giving it to sectarian causes). A governor or powerful congressman/regulator could demand all sorts of concessions from religious institutions or their funding could be withheld or reduced. In other words, it gives the government direct (and/or indirect) influence over the religion itself.