(The below is only my personal observations from infrequent exposure. If someone has more knowledge or experience, please correct me. Really, please do.)
> I had no idea some enjoyed so much autonomy, or that something other than the democratically elected government arranged for courts to exist in some regions.
I just want to clarify that while "autonomy" may be the case on paper, the reality on the ground often seems more bleak -- at least to me :(
From what I can see as an outsider living near and traveling through some communities in the West, it seems like they don't really "enjoy autonomy" as much as "learned to hang on, by a thread, to what they have left".
It's a really sad sight to behold, ancient cultures reduced to communities smaller and poorer than your average middle-class California suburb. Depending on the particular tribe and reservation, a lot of times they are more "trailers and mobile homes scattered over acrid plateaus" than "thriving communities". The cultural ties they have to the past -- in food, in dance, in song, in story, in way of life -- seem to be more acts of remembrance, occasionally celebratory but often somber or perhaps even mournful :( By and large the dominant theme today is poverty. Overwhelming poverty, at 300% that of the White poverty rate.
I don't mean that they are joyless or hopeless, but that it's a very far cry from what real sovereignty and community would look like if they still controlled their ancestral extents and resources, everything from land to water to fish & game and real diplomatic autonomy, not as a "domestic dependent nation" beholden to the US Congress. Today they exist not in equal partnership with the United States, but at its mercy.
Whatever US laws say on paper, in reality the tribes are often treated as second-class citizens living in the invisible margins of American life, and only in ways that Congress allows. And to the extent they are protected by US law at all, those laws can be changed on a whim by a new generation/administration of US policymakers, as they have many, many times in the past through the United States' frequent betrayals. Yes, the tribes have some specialized courts and law enforcement today, but that's more Congress throwing them a bone. It's not true sovereignty, but something more like house arrest, where they're "allowed" to manage some of their internal affairs, but only as long as and only in ways that Congress sees fit.
They don't often have their own economies beyond casinos and tourism, they can't manage their own foreign relations with other nations outside the USA, they often don't own their own land, etc. (https://en.wikipedia.org/wiki/Tribal_sovereignty_in_the_Unit...). TLDR the United States forcibly relocated the tribes to some of the worst geographies (after killing many), and then told them what they are and aren't allowed to do on the land that we allow them to live on. Many are freely murdered and never investigated (https://www.nativehope.org/missing-and-murdered-indigenous-w...).
> If I ever visit the USA, this sounds like they'll be some of the culturally most interesting places to include in the trip.
If you ever visit (welcome! sorry for the grimness above), just please approach them with the respect you'd show any unfamiliar culture. Some tribes have facilities (like museums or visitors' centers) for receiving tourists. Others may be less welcoming, especially, well, if you're a White European.
If you want to understand their cultures, don't just see them from the US side, with our laws and statistics and treaties glossing over the centuries of violence and erasures.
In books, movies, documentaries, and even video games, many tribes have tried to tell their own stories from their own points of view. If you want to visit any, it might be worth doing a bit of reading beforehand and then maybe reaching out to tribespeople to see if they would be willing to actually meet you...?
If you just drive near or through these communities, you'll only see them at their surface levels, because they're usually not a tourist destination in the way that, say, the welcome centers of a US state would be.
Sorry if any of the above sounded preachy. That wasn't my intent. I just didn't want you to blindly stumble into a situation thinking things are great now, with tribal members welcoming you with open arms and eager to show you their culture.
Anyway, I do hope you come visit someday and see for yourself, then come back and tell us where we were wrong!