(laughing) So much for my shooting off a post without thinking first!
It's been a while since my electronics classes, let's see if I remember this right...
The AC coming in is going to your transformer to step down to the 12V. But some will just be taken off to supply the AC switches. So the AC switches should work as long as you have continuity (which you do).
Coming out of your transformer you should still have AC, but a much smaller voltage. It would be good to test and make sure the output from the transformer is what it should be, I'm guessing in the 12-15V range. I don't know what voltage they would supply to the rectifier.
The bridge rectifier converts the AC to DC, so the output from that should be your 12V DC, although it will not be too smooth (stable). A rectifier is just a bunch of diodes, but check to make sure they are all hooked up.
The capacitor is most likely used to smooth out the 12V output from the rectifier.
I would carefully guess that the transformer is the culprit. (Is that green stuff on the transformer corrosion?) It may still be working, but a tap could have broken off, or a turn (many turns?) in the coil could have shorted somewhere. Unfortunately the transformer is the most expensive part of the power supply.
Sorry if you already know all this, I'm not trying to point out the obvious.