Shoes on a jumping horse is a case by case basis. I know some who are just fine without shoes, I know some who can barely canter without them. So you pulled his shoes and rode him around and he was perfectly sound? And you've been riding him and his feet have held up, he has enough traction, and he's still sound? If that's the case I doubt he's going to all of a sudden fall apart! If it was hurting him he would have let you know by now. if the chipping gets progressively worse, yes, I would wait to jump him but if it's just minor sounds like you're just fine going barefoot.
The hardest thing for a green horse to learn when they're learning how to jump is where to put their legs, how to maintain a steady rhythm, how to maintain a straight line, how to stay balanced in the corners, etc. All these things you can work on with really small fences. -if you're worried about jumping him bigger without shoes. Make them pretty scary and practice at a smaller height. Once he learns how to jump and is desensitized to scary things, raising the fences will not be a big deal at all.