I don't think Brannaman is doing English in western tack. Brannanan is doing classical, regardless of tack. His is the vaquero style, which in turn comes from the Spanish which in turn comes from old school European, be it French (de la Guerriniere), Italian (Neapolitan ) or Austrian(Vienna). Tack, and some of the training methods have been modified according to needs, be it war, what it was originally intended for, or work.
Oh, and Rollkur was nowhere to be seen. And certainly nobody would have had the idea of slapping a shanked bit or a curb into a horse's mouth just because it was 5 years old.
And yes, a well trained horse can and should be ridden in a plain snaffle up to the highest levels.
If Western Dressage, or whatever you want to call it, gets people on the right path, away from said peanut rolling, half dead horses, that would be really nice.