You are right about the fishing line having a loose end being an issue. Wish I could help you one that but I've never had to take that mech apart enough to see how it works....IIRC I did have one of those yokes off to de-cat a CRT, but I can't remember looking at the centering mech.
Good luck on it! If that set has a tube amp chassis, and was in a more modern cabinet it would be my ideal combo.