Did you have bloodwork done to check for Lyme and EPM, etc? I assume so, because I personally consider that to be part of a basic work up, but just in case…
My guy never got to the point of throwing people off, but he did get to the point of balking and threatening to buck when asked to canter left. If we were going left and I asked for a counter canter, he was fine. He was fine walking and trotting, he was fine cantering right. It was just cantering that one direction.
He had lyme disease. Even after treating the lyme, it took us a while to get his left lead canter back (partially because I’d learned bad habits), but we got there eventually.
Absolutely no guarantee this is your issue, but just in case it has not been explored, I’d check for Lyme.