So far I don't see a refutation of my arguement here. But, then, perhaps we disagree on what a "public square" is. To my mind, a public square is where the citizens can hold forth, using their First Amendment rights, to express their views on religion, the government, and the other issues of the day. If an actual square or park on common land, then it is maintained by the government. In other cases the public square can be said to be the newspapers, the radio and television airwaves, or even the Internet. In all cases, though, with a very few restrictions the government has no say in what is said there.
Whereas the purpose of the public schools is to provide a place for the teaching of reading, writing, arithmetic, history, geography, civics, etc. In contrast to the public square, what is said there is subject to the approval and control of government employees (the school's principal and teachers). The teaching of a particular religion, and leading the students in exercises pertaining to it, by government employees has no place there.
Which is not to say that religion cannot be mentioned in the public schools. The teaching of a course in comparative religions, for example, is no violation of this. The teaching of the historical role that religion has had in the shaping of America certainly belongs in an American History class, although as you have seen there are varying opinions on just what role religion had in this matter. Use of the school buildings for religious exercises after the school day, when the students have left and attendance is voluntary is accepted by the Supreme Court, as long as there is no preference given to any particular religions over others that wish to use the schools for this purpose. And anyone who says that there is no prayer in a school during the school day perhaps has not been in a school during finals week. Although, such prayer is generally silent, private, and not led by a government employee.
What's to refute Ron? Your argument that an accomodation paid for by the citizens of that town is not the public square? You torture the notion. If the citizens pay for it, then it is the public square subject to all manner of federal and state regulation. If it were a private club, you could certainly ban the word God, or women or white guys with big bellies. You can't.