Horses are 'creatures of habit'. I do not think that a circle as big as an arena puts more pressure on him going one way or the other at a walk. A tight circle would be a different matter. He has just 'talked' someone into going out that gate from that direction and thinks it should work again. He has gotten a 'pay-off' going that direction and he is just looking for another one (like leaving the arena and being put up).
Just like I make students go to the far corner of the arena, stop, stand still a while, dismount and lead their horses to the gate, you should do the same when you walk him. Get after him for stopping at the gate (if you can safely do it without him fighting and re-injuring himself) and then never let him turn and go out the gate when he is near it. If you get this horse sound enough to ride in there, he will do the same thing with a rider.