If you help him go forwards you will avoid any issues you might otherwise get from his muscles being used to go in circles (assuming they are tight circles). Just to even him out, you might want to take the weight for him after some initial circles and help him make more natural movements in the more natural direction.
I have seen many dogs missing that leg move forward, fall, move forward, fall, and use those falls to correct their direction, but I think your dog is too elderly for that. It is fantastic that he remains upright. Sounds like a good harness!
He has no issues moving forward. The faster he moves forward the straighter his course is. Momentum keeps him balanced. He needs no help from me in this regard. It is when he slows or stops that he leans to the right. I use the harness as a short term aid rather than prolonged. When he is having issues standing back up or crouching over to poop I provides some lift for balance or lack of strength.
Hannibal, the general not the cannibal, was very fit for most of his life so still retains a lot of his muscle mass. It is his joints and arthritis which are the issue. The strength training is more about adjusting him to the new stance and limited mobility then recovering his strength at his peak. He still trying to move like he is young. Sitting up on his hind legs. Standing up on the hind leg to poke me when he wants something. I am trying to teach him the dog version of "Old man you are going to break a hip!"
Maybe I will post a pic of my dog instead of a pic of me. His grey beard will put all to shame! Besides the ladies can not resist a dog with 3 legs. Trust me I know. I took him to the beach with me a
lot