What are we checking for? I view balance from the perspective of an alignment i.e. tripod. I am sure there are more complete definitions. I have also noticed that the only way I can stand on a crate (like Mr. Doyle does) is to keep my head between my feet. I used to work on a "steady head" , but it was was seldom a centered head. Balance is indeed essentially without it there is little chance of a centered arc for impact etc.
I'd say balance has much to do with the fundamental alignments of the pivot. You have multiple centers in the pivot working with the ultimate goal to keep the hands as close as possible to the selected plane and strike the ball with the proper relation to low point. Where the "multiple centers" are aligned at impact has implications on the path of the club and the face and the shaft location (face, head, shaft) . . . . so balance I'd say is way more than just staying "erect" (uh huh huhh uh huhh) . .. . there's an "axis" theoretically that were trying to pivot and move the move the club in relation to . .. . but with that hula hula motion of the hips, spine and shoulders there's a lot of stuff going on to make that happen and still hit the ball precisely while remaining "erect".