Can you elaborate further? Not saying you're wrong at all I just know nothing about westerns
Its kind like how Star Wars is not a science fiction film even though it has spaceship, lasers, and technology. It is a fantasy film.
The Matrix has wired-martial arts and philosophical undertones but its not a kung fu movie
Tombstone is an action/drama loosely based on wyatt earp who lived during those time periods of the old west
the setting, gunfights, all those are apart of westerns, but those are just decorations westerners are character driven, focusing more on the main character struggling with his principles morality and honor as they clash with society much like the samurai films they are derived from
that's why the main character is usually stoic with a dark past seeking a new life/path are usually dangerous but shy away from what they are capable of doing etc and they always come into conflict with the world around which is transitioning into a different era where their ideals and way of life are becoming a thing of the past.