You have a training problem. You need to do 2 - 10 times as much slow work and fast work. Horses 'burn out' badly on too much fast work.
You also need to pull your horse up and leave the arena a different way on practice sessions. You DO NOT need to teach a horse or practice running out of the gate.
Let me play this scenario for you. It may or may not apply in your case, but it shows the disconnect between what people think a horse is learning and what the horse is actually learning. Sometimes there is a world of difference between the two.
Many barrel horses are reluctant to go in the arena. Could it be because they go in the arena, are whipped and spurred around each barrel, have the reins jerks and pulled on at each barrel and then are whipped and spurred all of the way out of the gate.
When they get out of the gate, the reins are dropped, the whipping and spurring stop and the horse is petted on for doing a good job.
Now, lets look at it through the horse's eyes. Rather than teach him to run hard, check and turn, he is whipped and spurred all of the way. Now, he is again whipped with an 'over and under' or spurred all the way out of the gate. As soon as he gets out of the arena, all the whipping, spurring an jerking stops and he is rewarded for leaving the arena. Why on earth would he rather be in the arena than outside where he is petted and ridden on a loose rein. Of course, he dreads the arena.
If you watch the top barrel futurity riders, you will see trained horses that do not get whipped, spurred and jerked around. They are TRAINED. Many run in nothing more than a snaffle.
When barrel horses get pushed too hard, run too many fast runs, get too many hard runs out of the gate, etc, many get really sick of it and want to stay outside where they get pulled up and petted.
I have helped many riders with arena sour horses. Most (but not all) have come back pretty nicely. There are several things that you can do that help rehabilitate a blown up horse. The most important thing is to do a LOT of slow work in the arena. If you go in and trot a couple of patterns, stop the horse and rest at the 3rd barrel (if that is possible), go to the far corner of the arena, get off, loosen the girth and tie the horse up there for an hour.
Give the horse 'relief' in the arena and not just after it runs out of the gate. Horses figure out pretty quickly where they would rather be. It is up to the rider / trainer to make their lives more pleasant in that arena than outside.
Hope this helps. If this is not what you are doing wrong in this particular case, please do not take offense at it. I just know this is one of the problems with many of the horses and riders. Pain, of course, can be a reason why horses do not want to go in and run, but you seem to have addressed that already.