Religion was so ingrained that it was felt unnecessary to write it into various founding documents. Our founders were familiar with a state supported religion, and they did not duplicate it in our founding documents. They were in error, first time I ever said our founders were in error, as we see in hindsight by what has happened in our parents and our own lives.
Whether a definitive statement is in our documents or not, it is in the basis of morality and law for our country.
"Religion was so ingrained that it was felt unnecessary to write it into various founding documents."
With all due respect, that's a rather convenient conclusion. One can apply it to all sorts of things that aren't there - sort of a pre-Constitutional penumbra to complement the penumbrae the Supreme Court has found over the years.
That being said, religion was not so ingrained that it was simply there, like water to fish, to be assumed into the Constitution. Thomas Jefferson, at the very least, who was quite involved in the drafting of the Constitution, deeply questioned the role of religion in American political institutions and found that those institutions functioned best when religion was taken out of them. That is the tenor of the Virginia legislation: An Act of Establishing Religious Freedom, that Thomas Jefferson wrote.
Thus, while it is true that religion was deeply ingrained in the culture, it was not unquestioned and was not simply assumed into the Constitution. The precepts that were learned from religion might have been incorporated, but only to the extent actually written into it, and not further.