That's a really good point and one I've been thinking about as well as I also have a fantasy world that has a Native American like-group. Also, Wanik, do you have any advice on do them right?
Correcting this, "I also have a fantasy world that has a Native American like-group," would be my first piece of advice.
Natives are fucking THOUSANDS of ethnic groups that range from small bands of loosely allied hunter-gathers that share a similar language and spiritual practice to full on urban centered imperial powers that will split your face open with a macuahuitl. Sure, there are
some broad strokes that can be painted over regions, but the idea of one group can encompass the entirety of people that lived from the tippy top of Alaska down to the shores of Tierra del Fuego needs to burn a fiery death.
Like, me, I'm Mohawk and Mixtec.
My Mohawk peeps traditionally were horticulturalists from around the St. Lawrence valley and the Finger Lakes region. We chilled in large fortified villages of longhouses where entire extended families lived, and we'd move these villages about roughly every generation or so to keep from exhausting the soil. We were also largely democratic with a gift-centered economy and very little social stratification despite belonging to a complex political alliance with the other Huadenosaunee/Iroquois tribes (you might have heard of us).
Meanwhile down in southern Mexico (just north of the Maya lands) are my other people, the Mixtec. That side of my family use to live in a society that was much more stratified. We had nobles, artisans, and peasants that lived in a bunch of small city states that were so good at killing each other we couldn't bother to come together to fight those Mexica assholes when they came a conquering (AKA the Aztecs).
Now, like since this is suppose to resemble the Wild West, you're probably going to be focusing on the plans tribes and possibly those of the Great Basin and the Southwest. Even just keeping an eye on the plains, there were more than your typical teepee dwelling folks. Like the Pawnee, Hidasta, Arikara, and Mandan lived in villages made from big earthen mounds for instance. Never mind getting into the variety of social organizations, alliances, spirituality, and so on and so forth.
So if I was you, I'd be getting my google-fu on looking for books and articles. Also try and look into linguistic studies, cosmology, and folklore to really get a better sense of Indigenous folks.