Separation of church and state is not in the Constitution. The closest you will get is the first amendment, which prevents Congress from establishing a religion, or preventing the free practice of one.
I’m with you that religion should not be forced in schools, but you can’t use the Constitution to prevent this particular issue.
The first amendment was used to bar separate states from establishing religion as well.
Congress has the authority to hound out any religious activity grander than mere expression thereof from any institution which acts in its name or in localized substitution of it, that would include private schools.
It is entirely possible to read the text and case law of the establishment clause in a way that enables the US to stop just short of laïcite levels of choking religious activity in public.
Those ten commandment monuments could very well be made grounds to incarcerate the perpetrators the same as if they had tried to lead a crusader’s coup, because by publicly evangelizing they’re basically declaring that intention, that they refuse to accept a status quo in which people who do not share their beliefs are permitted to exist unaccosted and without declaration of judgement of their character for faith alone.
Plus really Christendom would thank the government for purging the public shouters. They’re literally called out in scripture as false believers who assemble loudly and publicly to make show of their displays of piety. So they can’t even argue sincere religious belief, because their own religion deems what they’re doing to be fake as all fuck and worth chasing out of society.
