I think that you are right that a TWH will most likely not be a great barrel horse. It is possible that he is fast, who knows?
I have a 24 year old TWH that my husband now rides. His canter is awesome, and he did perfect flying lead changes the first time he was asked. Never a cross fire, never a missed lead. However, he has a nice, slow, rocking horse canter. He only gets in a hurry if chased by an evil poodle or if someone's hat falls off. B.B. King once sang, "I am built for comfort, not for speed." Well that is my horse. Slow, comfortable ride. On that thought, maybe I should take him back from my husband........