Well, the Starr revolvers were probably not "the guns that won the West", but I am sure some were there, as just about every kind of firearm in use at that time "went West" with someone. They would probably have given good service, but would have been considered obsolete in the new metallic cartridge era, and I have never seen or heard of a cartridge conversion of one. Since the company went out of business in 1867, there would have been no spare parts available, another factor tending to discourage much use.
While we tend to think of the Colt SAA as THE western handgun, they were expensive (a month's pay for a cowboy) and there were probably a hundred cheap "suicide specials" in people's pockets for every SAA on a soldier's or lawman's hip. And hundreds of thousands of "in between" guns, reliable and with reasonable power, that never gained the near-worship accorded the big Colt.