Sometimes it's all about the "shooty, shooty, bang, bang"... ;-)

West of Loathing, of course!

Purple Dead?

Just a quick guess...

https://www.gog.com/en/game/rustler_grand_theft_horse
https://www.gog.com/en/game/dying_light_vintage_gunslinger_bundle (dlc)
https://www.gog.com/en/game/mad_max (not the wild west, but the wild apocalypse)

No. I daresay there aren't any (besides Gun) anywhere that aren't the Red Dead games. Westerado, if you don't mind retro graphics? Law of the West, if you don't mind especially old games...

If you've played GUN...

... you might want to check out Wild West Dynasty, but read the reviews and watch some recent gameplay videos.

Not open-world, but very good Wild West games IMO are...

... the Call of Juarez games (although I might skip #1).

There are very few American Wild West released and even fewer released on GOG.

Off-GOG you might check out This is My Land.

most has been said already if the craziness (along with the open world) of a game makes it up for you then
https://www.gog.com/en/game/saints_row_gat_out_of_hell
https://www.gog.com/en/game/dying_light_vintage_gunslinger_bundle
or maybe let's use gog's tags https://www.gog.com/en/games?tags=open-world,third-person
good luck! 🍀 🗝

Well, not only is GUN (I know, old hat) DRM-free, it's also............. not made by R*.
I should really just play it. Unsure if it'll be fine on Linux but why wouldn't it be with Wine as good as it is already.

It's very much like a Wild West comic book (similar to Red Dead Revolver), but the setting, mechanics, and open world are fun (albeit the open world is sparse). But if the open world isn't your main draw, I'd play the Call of Juarez games (especially Bound by Blood and Gunslinger). Love those games.