While sometimes it's fine to take jumping lessons on a horse that is point-and-shoot head in air auto-pilot lesson horse, in that case I would want to be able to work solely on me / my position / my balance. However if you're not working in 2pt regularly in lessons and the horse is a plodding/heavy jumper I wouldn't consider that auto-pilot so much as off balance like you said.
As far as auto-pilot those horses may be straight/stiff but I consider them still light on their feet. Perhaps not round, but a good solid lesson citizen where I don't have to worry about the horse so much as I can focus on what I am doing, my position, my balance, etc. it (unfortunately) doesn't sound like this horse fits that description.
I also find it odd that you are not doing more in 2pt. One of my favorite exercises is riding in 2pt and applying my dressage training to "jumps' where jumps start as ground poles then eventually become a low course. Not only does it do wonders for leg strength and balance, using my dressage position and foundation helps me bend the horse better and ride my corners, transitions, change of pace, etc., better that as the ground poles become fences they get easier and easier to ride because i'm riding basically dressage with speed bumps as my trainer used to say :)