A horse is a horse, of course, of course!
Gaited or no. Usually, gaited horses do just as well in "non-gaited sports" as other horses who aren't bred for that event. Heck, sometimes gaited breeds, like other "darkhorse" contenders in other sports, do surprisingly well.
My mare's sire was a WGC on the SSHBEA sport horse circuit as reiner, cutter, and team penner. He's a spotted saddle horse. I can't say very much else, since I'm not personally into Western-y sports.