The idea of the Declaration of Independence as a mission statement originated not with the Straussians, but with black abolitionists during the Revolution. In general, no one bought the idea at the time; indeed, you will search in vain for anyone having said that during the ratification campaigns.
As to the absence of the idea that power comes from the people from the Constitution, I suggest that you re-read Articles V and VII with the fact that “states” refers to “the sovereign people of each state” in mind. Then you’ll see that the Constitution is actually based on this idea.
