You are hunkering down.<quoted text>
That is not a foundational or fundamental concept - one upon which others rest or depend, as a house rests on its foundation. Foundational concepts underlying the US Constitution include things like limited government and individual political freedom. Other ideas derive from these.
For example, from the foundational idea of limited government derives secondary ideas like separation of powers, checks and balances, transparency in government, enumeration of specific governmental powers.
And from the foundational idea of individual freedom derives secondary ideas such as freedom of expression, right to privacy, and right to a speedy trial by peers.
Attributing rights to a god is not the foundation of anything in the Constitution or American government. If that were a foundational concept, there would be provisions to determine which rights that god had given, and there would be safeguards built into the Constitution to prevent us from changing those rights.
The fact that the Founders came up with the Bill of Rights on their own rather than looking to their bibles for the rights granted there, and that they built ways to add to that list as their descendants came up with new ideas, attests to the fact that they understood that men were the source of those rights. The god mention is lip service, not a fundamental concept.
You could say that those men believed that their rights were endowed by Mickey Mouse, and it would be no more or less meaningful - no more or less foundational - than what you wrote instead. No fewer ideas derive from that belief than one citing a god.
You can say that the U.S. government was founded by people WITH an incidental belief that rights are endowed by God, but there it is incorrect to say that it was founded ON that belief.
Individual freedom is a secondary concept to creation by God. Then other rights are secondary to that.
That intent is as clear as ink on paper can make it:
"... to assume among the powers of the earth, the separate and equal station to which the Laws of Nature and of Nature's God entitle them,..."
Regarding items like separation of powers, checks and balances - these were derived directly from a religious view that the unchecked heart of man, and thus his government, would move toward power and subjugation of others. They referred to it as "the depravity of man". The most cited authority on this was Montesquieu, who endorsed 3 separate but equal branches. He based it on Jeremiah 17:9.
Incidentally, you mentioned a "right to privacy". The founders considered no such right, and did not include it anywhere.