I don't think I'd go that far
Jackson is definitely a jumper at heart, I don't think I could ever give it up competitively.
Half-halts are one of those things that I say I understand, but just can't (yet) find a way to explain them. But that makes sense, adjustability and accessibility seem to go hand in hand, and I can definitely see what you're saying.
So back-tracking a bit, then, to what Tiny brought up (connection). Jackson seems to have this deep hatred for single joints, his previous owner rode him in a single-joint snaffle and a nice, tight flash. Our progression of bits has gone from single joint, french link, back to a single joint (I'll have to post a picture, it was a 'training' snaffle by Herm Sprenger, I really liked it and he seemed to like it a lot more than the first snaffle) and now we've switched to a waterford / bubble bit, which I was a bit weary of, but he seems a LOT more comfortable with more joints in his mouth, and it isn't pinching, so I'll leave it. We've also removed the flash, and he's happy to no longer evade the bit 90% of the time. On the rare occasion that he does toss his head in the air, it's easily corrected, however he's not 'on the bit' 100% of the time which honestly is something I'm not sure how to address.
We work on pushing him forward through seat and leg, and keeping a contact, pushing him forward into the contact and 'playing' with the inside rein. I don't want to say I don't trust my coaches instruction, but I wonder if the reliance on the inside rein is technically 'correct' or if it's just an attempt to encourage a false headset. Especially for a horse who doesn't have the best relationship with contact in the past, recommendations? Exercises?