Doors – paint a door without removing it from the hinges

If the existing setup works, I don't see any reason why it wouldn't work with new doors instead. One down side to this setup though, is that doors without jambs aren't that secure and could probably be kicked in fairly easily. If it were my garage, and I didn't want a standard fold up door or a roll-up, I'd probably put in 2 larger barn style doors, or if you have the space, one of those sliding barn style doors that has an overhead track to slide on.

You say

The new coats of paint are not too thick so it's not the paint job.

and this is a common misconception. Yes, it looks like the paint is negligibly thin but it actually adds up. When a door is constructed it has size that allow for painting it once, maybe twice, and then a new layer of paint will prevent it from closing.

You should have cleaned the previous paint off the door edges.