If you want to be like that about, what we think of as American "cowboy" culture only lasted 25-years or so, Civil War-ish to 1880-ish. Yet we treat it like it was a 100-year block of Western history.
To be fair, I’d say cowboy appearances would be relatively proportional to the population, maybe 1 or 2% of each series…
Except DS9, which has a bit of an Alamo obsession.
Maybe not literally, but the season 3 episode where Discovery arrived in the future went hard on the western vibes. I think they even included swinging saloon doors at one point.