My theory is that birds that usually like to be on their backs are those that will wrestle when they play like caiques and conures(?). I don't think that's really to the cockatiel's natural strengths.
I think harness training or not is up to the person, but I believe that cockatiels can be happy and exercised if they have a safe room in the house. If the point is to get the bird fresh air and sunshine then a sturdy travel cage or carrier is an option. Everyone's bird and situation is different, but I personally would not free fly my cockatiels. I figure if falconers lose their kestrels (cockatiel-sized falcon) to larger birds of prey, then what hope would a cockatiel have? We have a lot of red tail and Cooper's hawks in my area.
Lolipop, it's great to hear things are improving with you and your bird. Take things at the bird's pace. It might be this bird just doesn't have the personality for the type of training you wish to do, and it's OK to feel a little disappointed! This sort of thing happens with all sorts of animals. And who knows, maybe after a year or two of bonding things might change.
