+1. I look at this way. If it had happened on ball 3, the game would be over. Why does ball 4 change that. If the winning run was forced in by the walk then I see it, but as it is I still say game over.
By your logic, if the same pitch were a strike 3 the run would score even if the defense subsequently put the BR out before he touched 1B. I mean the R3 scored because of the wild pitch, not because the batter swung at it.
If it were Ball 3, the game would indeed be over. However, since it was Ball 4, the batter became a runner and the "play" isn't over until the BR touches 1B. If he fails to, he is properly called out and the run nullified per the 4.09(a) Exception.
The difference, as you pointed out, was that the batter became a batter-runner on the pitch that was Ball 4. He has responsibilities, one of which includes not being put out before reaching first base. If he does, no runs can score.
Those that are saying that the run didn't score on the walk but "on a wild pitch" are using the same argument coaches use when saying first base isn't occupied on a third strike because the runner was stealing. Sounds good, but has no basis in the rules.