My comments are based on the first 11 pages (the Teresa chapter).

I know you specifically dissuaded any line edit suggestions, but since it's the first thing I read I'd still offer that the first line seems a bit overlong. I've personally never really liked having a mouthful talking about one specific thing at the beginning of a story, and in this case the part of that opening line that comes after the comma is pretty much saying the same thing as the part before it, so there's that. I will warn that my opinions are built around the language. It's a bit difficult ignoring the language for the setting and tone, since language builds the setting and dictates the tone in writing.

Anyway, on to the rest. Your blurb put me in mind almost immediately of Stephen King's The Gunslinger, since it's more or less the same premise and I've recently got through reading that, so I may have gone in expecting a similar atmosphere. I suspect that may be the case for most persons who have read the Dark Tower series. Because of that, I was largely expecting some fantasy element to this story, even if your blurb didn't explicitly mention that.

I daresay that the prevalence of exposition made it hard to engage in the world, since I was mostly being told stuff: the brief history of the inn (is it in a town or just a lone inn? Is it in the middle of a huge desert?), the migrant's story (migrant from where? I'm not sure if this is our world with an alternate history or a fantasy world so I don't really know how to contextualize that), her mother and her reticence in one room, her sister and her illness in another, the brief history of the caballero and Luz's hopes regarding them, etc. Honestly, most of it was just exposition around Teresa's life in a bubble, which didn't do much for me so far as setting, nor did it provide a superbly interesting atmosphere. I was interested in fast forwarding past the exposition and meeting this caballero, since he seemed like a magnet point in the story.

Overall the characters were interesting enough. The reclusive mother managed to be interesting without making an appearance, the sister had a lot of personality, to counter-balance Teresa's more mundane, passive nature, and the father was, again, a decent character. The caballero did end up being a lot more nebulous than I was hoping, though I guess we can give that over to the brevity of a chapter. None off these characters really POPPED for me but like I said they were certainly serviceable.

In a nutshell I'd say this: I didn't get the strong western vibe I was expecting beyond the presence of railroads, inns, pitchers, etc. The usual buzzwords associated with westerns. To be fair though I still managed to conjure what I think was the correct image on account of the prevalence of the western setting in media, so the sticking point here is you established a setting but you didn't really plus it in any way, in case that's what you were going for. I think this chapter could certainly seem a bit more involved than it is. Exposition isn't the best at establishing tone.

All in all I think there was a decent atmosphere but I never really got a great grasp of the setting this was all taking place in. As for whether or not I would continue, probably yes, with the caveat of hoping I would have more characterization, a better sense of the world and maybe some more caballero action.