Barrel racing is ALL about control. Bottom line...sure, any horse can go in there and run the pattern. That doesnt mean they're going to execute a proper turn or that they are going through willingly and attentively.
Like most other diciplines, a barrel horse requires lots and lots of control. Lateral movements,collection, and softness. The twist? They have to be able to maintain everything in a gallop. A barrel horse has to be smart enough to learn and progress, but dumb enough not to out smart the rider.
Like you said origionally you start slow on the pattern. (after already having all the basics down) Starting at a walk. There are a hundred different ways to put a horse on a pattern. It all depends on the horse itself. Some horses need a simpler start and some can handle a more technical ride. (meaning adding right off the bat some lateral movements)
Once a horse is able to maneuver calmly and correctly around the pattern at a walk, you move up to the trot...so on and so forth.
Ironically the KEY to a good reliable barrel horse is lots and lots and tons and tons of SLOW work. Limit pattern work to only a couple times a week. Less if the horse gets frazzled easily. Focus on suppling the horse OFF the pattern. Its harder to keep a barrel horse fresh and calm than to teach them the pattern in the first place.
Remember that anyone can hop on a horse a yeehaw through a pattern...but it takes a true horseman to be called a barrel racer.