How does training have anything to do with it? A quarter horse is a quarter horse for a reason...they are more athletic in certain things. Obviously you can't train a QH that is 14.2hh to outjump a natural 17hh TB, just like you can't train a 17hh TB to turn a barrel faster than a 14.2hh QH. That just doesn't make sense! Someone correct me if I'm wrong!
A QH is a QH for a reason, they are the fastest horse in the quarter mile while a TB with a longer stride is the best endurance horse. But a QH will smoke a TB in the quarter mile. Quarter horses are the fastest horses top speed wise. Therefore in something like barrel racing where its quick sprints, they should be the fastest horse for the sport IMO.