The Declaration of Independence is the “Why?”
The United States Constitution is the “How?”
The “Why?” tells us what they were thinking in the “How?”
To read the Declaration of Independence and conclude that the framers wanted God removed from any government entity shows how intellectually dishonest they are.
They already knew that there would be various sorts of expressions of faith, and openly participated in the same. The one verboten thing was to designate some particular denomination as being the US equivalent of a Church of England. The promise not to do this, allowed for there to be peace between what might otherwise be various denominations squabbling for power.
The modern jurisprudence on this matter twists legal philosophy into a shape that was not envisioned by any founder.