Scootz, you're generally on the right track in coaching him. Here's a few hints - some of which you already know ...
His last step is too long. His penultimate is OK, but then he strides out - putting him under.
His plant isn't high enough, and isn't forwards enough. When the pole hits, he's jerked off the ground because he hasn't stretched his top arm up as high as possible. His elbow is bent when the pole hits. This is quite common with vaulters that haven't done much highbar work. The more he swings on a highbar, the more comfortable he'll feel with his top arm fully stretched.
Reason: It's pretty hard NOT to have his arms stretched when he swings on a highbar!
As you know, he isn't jumping on takeoff. Lead knee needs to be driven aggressively forwards and upwards - like doing a layup.
Have him try dunking on a rim set at maybe 8-9 feet. Use a ball that he can palm. To start with, don't try to give him any hints - just tell him to dunk, and raise the rim high enough that he MUST jump on takeoff to dunk.
Once you determine how high he can dunk on his own, then start coaching him on how to jump higher - with a higher rim. e.g. driving the lead knee up, shortening the last step, etc.
His gut muscles are weak. Too weak to even do a trail leg swing with his leg straight. Get him on a highbar and teach him how to swing. If he can't do it on a highbar, then he can't be expected to do it on the pole.
The stronger his gut muscles get, the smoother his swing will be. Sorry - no magic here - it will take a few months before his gut muscles get strong enough to swing properly.
If he's not a senior, then better luck next year - get him doing highbar and rings (and even simple situps) all thru the off-season. He'll have a much better swing next year.
I don't know if you realize this or not, but you can use the left and right arrow keys on your PC to step frame-by-frame thru his plant, takeoff, and swing. If you do that, you'll notice his bent arm when the pole hits, and everything after that is due to his poor plant (and not jumping).
Try getting him to stretch the top hand up as he's jumping up - just like dunking a ball. The impulse of stretching his top hand up should be quite similar to the impulse of stretching up to dunk a ball. And of course, he'll never dunk the ball if he doesn't jump off the ground.
Here's an extra drill to try - kinda fun: Have him dunk a ball and stretch into the C position at the same time!
This is a more advanced drill, and should not be attempted until he has the basic jumping technique down pat in his normal dunks. The landing from this drill should be back onto his takeoff foot. Don't take this drill too seriously - it's just for fun and doesn't emulate the takeoff to the C technique perfectly - but it's close.
Another drill to try - if you have rings that can be raised or lowered - is to have him do a runup and takeoff, then grab the rings. The idea here is to jump high off the ground and stretch the arms up high, ending up in the stretched C. This drill is closer to a PV takeoff than the dunking drills. Another advantage of this drill is that once he grabs onto the rings, he can then swing out of the C. Actually all the way up to the Inverted I position.
I like the rings better than the highbar for this drill, because the bar tends to get in the way on the highbar. So rings simulates the vault better for this run-and-jump drill. But for most other gym drills, I prefer highbar to rings.
Kirk