I have never tried barrels before.. but here is what I am seeing :
See how his head is up? That is your fault. He wants his head. If you have tight reins so he doesn't go too fast.. that I don't get. If he can't walk around the arena on a loose rein you can NOT be galloping him around barrels!
You are fairly unbalanced and flapping around a lot.
When he was rearing in the beginning, that wasn't him. You obviously need to walk/trot this horse around a little before starting barrels and not expect him to be doing all these fancy turns and yada. If he doesn't want to go, don't force him!! It's a matter of he doesn't want to be ridden and that's not his fault.
You need to work on your balance and you NEED to retrain this horse before you hurt him or yourself!!
His gaits seemed very unbalanced, too.