It still could be that some other component shorted. You can't dissipate power into a short, as there is no resistance in a short, so the load is reflected to the source (xfmr), which suffers. If you have an ohm meter, look for a short. You may have to lift the wires on the xfmr to do this. Typically a capacitor, but maybe a diode bridge is dead. Old caps are usually the problem. Is there a fuse hiding in there somewhere?
By the size of the xfmr, you can tell about how much power it is. Probably 10 to 15W, but I can't tell. There were a lot of 12.6V xfmrs used in those times. If you lift the wires on the xfmr, then ohm the windings to see if they are still good. If they are compare the resistance of the primary to the secondary. This tells you the voltage. It should be replaced regardless.