I can't see the video but I have to first comment about the side reins.
Putting them on the tightest hole will not help him to relax, which is the opposite of what he's doing when his head is up in the air.
Take the side reins off, they aren't being used properly..
Second, I will comment on what I think is going on, due to lack of ability to see the video. He's not using his back from the sound of it which means he's hollow and probably not balanced either. And could be in pain. First, check to make sure he isn't in pain. I'd check with a different vet and find a chiropractor or even a equine massage person. Check the saddle fit too. Check for feet bruising or underlying problems.
After everything has been checked, I recommend you really need to go back to absolute basics with him.
My advice? He needs to go back to walking and learn to relax and work with his back and work with the rider. Don't even trot him until his walk is perfect. Even if it bores you to tears, and your horse to tears, as soon as he begins to understand how to use his back, consider that a good note and end on it. Build on that, until he can do it walking comfortably, then move up to the trot transition. Don't keep trotting if he throws his head and loses his back muscle. Back down to a walk, get him prepared, then trot transition again. If he lasts, good! Do a few steps and then back to walk. End on a good note.
Once he can trot with his back, without being hollow, then you can do the same for canter. Transition using back. When he stops using his back, back down to a trot, and if he's wound up at the trot, go to a walk. Prepare.. back to a trot, if he's good, back to a canter transition. If he handled it, continue then (BEFORE he falls apart) go back to a trot.
While you're doing these gaits, incorporate figures every so often to change things up, lateral movements if he knows them. Get him relaxed before you switch things up.
End on a good note. Yes this will be frustrating. Don't try to do it all in one lesson/session. Break it up into little sessions. Help him relax and engage that back. The head is part of using his back in such that when he uses his back, he will be relaxed and his head will not brace upwards.
I'm not an expert, but that's how I would do it.
Good luck. That's my advice.