I don't see having contact on the bit as pulling on the horse. Plenty of discipline's have a nice steady contact, and even ones that don't are based off that same contact/connection (and simply train it in other ways so the reins are longer). I think there's a lot being assumed here by people who may not ride the same way. I'm not sure why there are so many assumptions going on tbh. I think it's pretty normal for someone to move around a little when they go from a very slow smooth horse to a big mover, has no one ever made that jump??
I completely agree with having private lessons. I didn't realize you were not. Even if you don't want to do it long term I would at least try to schedule one or two while you're getting used to him. It will make your job 10x more difficult to try to get used to him in a group setting.