[Arran]

I can see that being a problem with the grounded filament. That is a lot of voltage between the heater and cathode, and a direct B+ short to ground wouldn't be kind to the trans. I didn't see the whole schematic, but if the filament weren't grounded, wouldn't that eliminate the problem? Well, maybe, maybe not ... it could then put that across the HK of the weaker signal tubes, then one of those short, = lots of excitement.   I have several radios that use this tube, but they all use the GT tube, not the G tube. One did have a bad power transformer - worked ok, but just ran super hot. Gradually got worse til I replaced it. Maybe was related to a failed 6X5 at some point??  But, even if one of the silicon diodes shorts, that wouldn't be any too kind to the trans either.

 I also have several radios that use a 6X5 rectifier, and not one set that I have encountered with this tube has had a failed power transformer, but I have run into at least two or more sets equiped with a 5Y4 that had a shorted high voltage winding, shorted filter caps or a shorted tone correction cap is all it takes. I also do not trust rectifier diodes, least of all the 1N4007 cheapies many seem to use, I figure it's better to pay the extra $.10 and use at least 3 amp rated diodes connected in series.
Regards
Arran