As recounted in a brief biography on the website for the American Quarter Horse Hall of Fame, into which he was inducted in 2004, Buster Welch specializes in horse cutting. In short, this is a sport in which competitors demonstrate their ability to herd cattle from horseback (via Horse Canada). Welch won first place in the National Horse Cutting Association World Championships Futurity competition four times in total. Furthermore, among other locations, he worked at the 6666 Ranch at one point in his career. Welch, in fact, is referenced on the 6666 Ranch website as the rider of a horse named Peppy San Badger, with whom he won a National Cutting Horse Association Futurity competition, and whose bloodline, by their estimation, is the second-best in the history of ranch horses.
Following his "Yellowstone" cameo, the 6666 Ranch foreman tells Jimmy that "there's three gods in Texas: the Almighty himself, Buster Welch, and George Strait." While this may be hyperbole, Welch is nevertheless a significant figure in his profession, easily recognizable to viewers familiar with the sport of horse cutting.