Just to clarify from the start - I am not a barrel racer.
However I AM a horse person
Seems to me that barrel racers (and their trusty steeds) get somewhat of a bad name in regards to having 'hot' or 'crazy' horses. Whilst there are probably some out there that perpetuate this stereotype, there are many that don't.
Although barrel racing and other competitve gaming endeavours are high energy and fast paced, not all horses will be uncontrollable as a result. Just look to the captain of the ship if you want answers as to how the horse is behaving prior, during and after the event. I am quite sure that no barrel racer wants a horse that is so above itself before entering the arena that they lose precious seconds while the horse jumps around like an idiot.
So my advice would be if you are looking at buying a barrel horse - have a good chat to the owner/rider. Find out how they handle the horse, what they expect of the horse in terms of behaviour and you will have a much better idea of what to expect from the horse in question.