The Western is a genre and like all genres it's got its own tropes and stock characters and even in the 19th century it never really adhered to what we might call realism.
You're very right, and that's part of where my discomfort comes in. The 19th century was the most blatantly genocidal against Native Americans (I'm not going to get into the modern genocide, that's a discussion for another forum), and there's plenty of racism and sexism extant in the historical record. But there were always, like
@doctorbadwolf and
@MGibster points out, exceptions to the norm. Especially on the fringes of colonial society, like the Western frontier.
But in the fictionalized version of that frontier, the "western", all that diversity and leeway between the societal expectations and the reality are flattened in favor of white male power fantasies. Women, especially white women, are either pure maidens to be rescued or "the harlot with a heart of gold". Brown people are either villainous, stupid, or we get the "magic brown person" or "noble savage" trope. Some of that is 19th century colonial propaganda, but it was also adopted wholeheartedly and uncritically into the modern western genre. I like hero stories as much as the next nerd, but I have a lot of trouble enjoying a genre like the "classic" western when everyone in it who looks like me is a racist caricature and target practice.
While modern writers and artists can and have begun to reclaim that diversity of storytelling in the western genre, most of my favorite examples have been done by way of the fantastical western, or the weird western, rather than a "classic" western.
I have some thoughts on why the western genre becomes more popular in the US during times when movements for social justice are also big in the public consciousness, and I think it boils down to the western as a form of myth-making for white people in the States. It's one of the ways in which the dominant culture asserts its right to rule, since our culture (such as it is) doesn't generally subscribe to the divine right of kings or other aristocracy. It also ties back in to a lot of other colonial claptrap about the Americas as an empty paradise for the white man, but I'm not going to go into depth on that here.
But all of that is part of why I feel so uncomfortable with using the western right out of the box. The entire foundation of the genre is based on some deeply racist and sexist ideology, and I just don't feel like replicating that in my free time without significant revision.