If you are a solid rider using the horn to really sit deep around the back of the barrel, holding the horn can be useful. It's more a personal thing. Unfortunatly, many people end up using it for balance. THey can not stay correct in the saddle without it. Thats where you get in trouble.
Yes yes yes you should be able to collect your barrel horse. You need to have every button on him possible. Take dressage lessons. You'll see a cut in your times. The more balanced your horse, the more balanced your turns, the faster the times. when you rate your asking your horse to collect over his haunces, pay attention and turn. If your horse is unbalanced, you will have no curve in your turn, dropped shoulder, knocked barrels, etc.