I am not a lawyer. And I think you are a lot more level headed than myself. But allow me to clear this up a bit.
Your initial point is so on the mark, That I assumed the OP was only referring to the Declaration of Independence and The Constitution. Not the bill of rights. Those sights are transient and cand never really get the legitimacy of natural law.
The Declaration of Independence lists ‘life, liberty, and the pursuit of happiness’ as unalienable rights. Those can never be taken away and are only in God’s hands. And His Natural Law supersedes all other law systems.