I should have the disclaimer in most posts: I'm not a trainer!!
That said, I don't believe letting a twh trot will "ruin" them. They'll not keel over and die, suffer from mental disorders or need "therapy"
I do believe in "form to function". If the op is looking for a barrel horse, then I'd agree it's best left for a horse that's most suited for the job. Have people barrel raced their tw's? Sure. Pole bending?? Sure. Is it the best horse for the job and good mechanically for the horse, well, not in my opinion. For those of you with more experience than I, please chime in.
When running barrels I think you want the fastest score and cleanest run. Most tw's cross fire in the canter, at least the ones that I've seen. I think it's due to them being so open in the stifle. That said, it's a bumpy uncomfortable ride and forget about being a smooth tight run around the barrel. I'd think it would put too much torque on the hocks and stifle and could cause more problems than it's worth down the road. This doesn't mean that a tw "can't" canter or will always cross fire and never have a clean canter.
I'm still learning so if I'm really off base, let me know.