Obviously the Constitution’s declaration that “we the people” form the government (rather than God, which had always been the case previously) is an example of the people leaving God. The Constitution itself is only a written document that reflects the beliefs of the people who voted for it. As a founding document though, it’s what people today look back at not the people behind it.
Many Kings, some of whom were the most abhorrent people to grace the face of the earth, were given great coronations in the name of God! Did that make their Kingdom’s Godly?
The Bible is just a book, until we the people put the Word, within, into action. I look at the U. S. Constitution in the exact same manner.
Putting the word “God” on, or in, something doesn’t make it Godly. You obviously think differently.