They agreed that the federal government should be TOTALLY impartial ~ and did that with language that strongly suggests that the federal government is simply not empowered to say something is or is not religious in nature.
The mistake is believe the Founders "classified religions" ~ they may well have done that in their spare time, but they didn't do it in the Constitution.
To mention religion in a document having legal force means that the question of what is religion is going to come up sooner or later. There’s no escaping the question. It’s not like asking what is the sun or what is the moon.
I believe that Ben Franklin crafted 5 fundamental “religious” ideas that he considered universal and that could be used in all public functions....like No. 1. There is one God....Two: That the ten commandments were valid for the fundamentals of our laws.....(that is why you find carvings of Moses and the ten Commandments on Federal buildings). That a day needed to be set aside for worship. I am not home so can’t find the quote of Franklin’s but I have read it and he applied it so there would be no religious arguments. The Founders understood the need for religious freedom in the public square.
He was underlying the religious foundations of this country, because this culture was based on Christian philosophy which was aligned with Natural Law Theory (John Locke)—Christianity is the only reasoned and logical religion according to Dinesh D’Souza and he makes a great argument for it in his book What makes Christianity so Great..