The Quakers were the North American pilots of a peaceful, tolerant, enlightened society, and with a few horrible exceptions, managed to peacefully coexist with the native Americans for large periods of time. How much of an impact did they have on the United States?

I can't speak to the general influence of Quakers on the United States, but I do want to point out this past post of mine (with follow-up from u/USReligionScholar) on Quaker support of Indian residential schools. Their own post in the thread will be of interest to you too. To learn more about residential schools in general, please see this post by u/EdHistory101.