When surveyed -- I believe in the early 90s -- even Western historians couldn't agree on the boundaries of the region. (Many respondents were not even internally consistent in their definitions.)
Of course, the Western genre, at least in any useful sense, has never been as strongly linked to rigid geographic boundaries as you are suggesting (especially on film).
One difficulty the historians would have is that part of the notion of the "West" means the frontier, implicitly opposed to the settled, even urban, East. In the South, with its endemic backwardness, it is indeed hard to find a boundary between the cultured regions and the wild ones. And in all of the country, the
frontier moved through time.
As for the Western genre, especially on film, being strongly linked to anything other than the setting, the difficulty is that, unlike the mystery or the romance or generational saga, there is no Western genre: There is no particular kind of story that is a "Western." What are called Westerns are in fact a wide diversity of genre stories and pretty much all that links them is a setting in what the perpetrators choose to call the "West."
As for those works that choose to incorporate motifs or references from previous "Westerns," I suppose you could
choose to dub them Westerns, and quit thinking about them. But this wouldn't really tell us anything about the real nature of the story. It would be like saying that The Jetsons was scifi, while ignoring the fact that it is actually the same genre, same kind of story, as The Honeymooners.
In any event, you aren't really making an argument but a quibble. Unless you have some thoughts on what kind of story a Western is, a sketch of the very basic kind of story structure found in the alleged "Westerm" genre? I personally have my own ideas on what might constitute a kind of core, a kind of hard Western that relates to the rest vaguely like the rest of SF relates to hard SF. But surely you would find yours more interesting. (And you would be surprised at how interested I would be in reading about them too.)
The modern Hollywood usage for "Western" does seem to be code for "hicks flicks." I'm sure you could do better than that.