I think they all do. This is a Broad category, like “War Films”. Shane is a cultural icon (and took an actor who would have been associated with Noir and put him in a completely different place in popular memory). Lst of the Mohicans is perfectly within the genre (and a great film). It deals with the Frontier. Perfectly a Western IMO. Westerns don’t have to take place post-Civil War in Arizona, for instance.
This is a Western, takes place in the Pampas: