I agree with everything you wrote, except the assumption that The Founding Fathers (I prefer Framers of the Constitution) are monolithic in thier thought. As individuals, I suspect there was some variation in thier ideas.
They were not uniform in their beliefs, but this particular piece of claptrap has been misused by so many left wingers that I felt it deserved a definitive reply. One merely has to read the book "The Constitutional Convention" by Boorstine, I think, to understand how disunited they were in their approach to government.
Having said that, the large majority of the Founding Fathers were devout Christians and any representation to the contrary is absolute (again) claptrap.
You may wish to read "The Fourth Great Awakening" by Fogel to understand the defining role Christianity played in our founding. It will leave you stunned.