There is a "remote" possibility the 24V transformer has a built in thermal fuse. I've had on fail on me before. They are on the primary, and taped inside the yellow tape that goes around the primary winding. Theory is, if the transformer overheats, it cuts out, permanently. However, those fuses also seem to fail with age. Options are to either replace the transformer, or carefully unwrap the primary tape to find that thermal fuse. No guarantee your transformer has one however. WHat are the ohm readings on that transformer, both primary and secondary. Any blown winding will be infinite resistance.
For your relays to click, there has to be some voltage feeding them, from somewhere. See if you can find the 24VDC source for the relays on the board itself. Trace that back to verify which transformer is supplying that 24V. I suspect the AC is being rectified on the board itself, so once you trace back from relay to the diode, it will be AC voltages backwards from there. It must connect via one of those connectors.