[Arran]

If the output transformer is the original one, and has continuity, it is probably fine. The low volume could be explained by the fact that the set has a field coil speaker, which will still pass sound without the field coil energized, but it will not be as loud, the field coil is powered directly from the 6.3 volt supply along with the tube heaters. Also the output transformer primary normally would have 150 or more D.C volts across it, the output of the signal generator will be much less, however you could connect it to the secondary side and see what the volume level is there. There is also a choke (Part #65) in series with the output transformer primary, so depending on what point you measured from it may test higher.
Regards
Arran