Unless you're playing cowboy games or just really want to look the part, why handicap yourself with an obsolete design just because you'll be in the saddle?
I don't ride much these days, but I never felt the need to go full dress. Yeah, I wore Wranglers and boots-which I always do. But otherwise, usually a Tee-shirt and ball cap, and my S&W 1006 or Glock 20 on my hip.
Not that I dislike single action revolvers or lever action rifles, but I was never into CAS, and I'd much rather have the 18 rounds of 10mm and a quick mag change available than my Beretta Stampede or Vaquero .44 Mag if I ran into trouble in the back country.
And I would have no qualms about carrying an AR on horseback, either, but would probably still choose a lever gun simply because ARs (or other magazine fed autoloaders with protruding grips) don't play nice with scabbards.