The French Revolution was also based on that, and in particular the Declaration of the Rights of Man and of the Citizen.
The reason it was more bloody is not only did they not actually respect it in practice, they also didn't have a coherent theory of government.
Moreover, they actually overthrew the existing government in its entirety.
In the United States, the predominant form of government at the time was government at the state level, and the states, starting as colonies, already had foundational documents in the form of colonial charters and, ultimately, state constitutions modeled on the federal constitution (although many states took quite a long time to do this and still operated under their charters for years or even decades after).
The state governments were not overthrown by the revolutionaries, who merely cast aside the foreign power which was not respecting their rights.
So after the French Revolution, there was nothing but the revolutionaries. After the American Revolution, the state governments still existed, still did the vast majority of governing, and the new federal government was deliberately limited in power.
The guillotine was a more humane way of massacring large amounts of your own people, but Americans have never been a huge fan of doing that, so we had no real need for guillotines.
We simply shot at the bad guys until they went back home and then went about our business.