[Arran]

I've never had an output transformer become shorted, the failure mode is usually an open primary, or in a center tapped one, one leg of the primary could become open whilst the other may still show continuity. The set will still play like this, but not as loud, and possibly with more distortion, I don't think it would cause overheating as the other output tube is basically removed from the circuit. Now a leaky tone correction cap can cause major issues, this is usually why I clip them out when powering up a set for the first time, in some cases they can cause a dead short across the B+.
Regards
Arran