Can someone explain to me how branding keeps any horse from being sold to kill buyers at an auction?
Brands are really hard to trace (just check any of the threads on HF of people trying to find out where their horses' brands came from... I don't think I've seen a successful one yet) and to my knowledge auctioneers are not required to make any effort to compare the horses at auction to any list of reported stolen horses. Even if you were able to successfully trace a brand, it only tells you who registered that brand, not who currently owns the horse.
I really think microchipping is a better alternative. The microchip is a unique identifier that can positively identify a specific horse and its registration can be easily updated to reflect changing ownership. Some breed registries already require them, and The Jockey Club recently announced they're going to start requiring them in 2017. If the other larger registries like AQHA, APHA, AHA, etc also follow suit it will make sense to start pressuring auction houses to scan for microchips.
“The horse is a mirror to your soul. Sometimes you might not like what you see. Sometimes you will.” - Buck Brannaman
"Nothing forced can ever be beautiful." - Xenophon