[Arran]

If you can solder/unsolder, use a VOM, and read a circuit diagram you can probably restore the set. The "Mystery Control" works by a pulse system similar to a rotary dial telephone except that it uses a radio transmitter instead of wires. You could build a home brew version just as an exercise but I don't think that the 1939-41 remote boxes are that hard to find since the radios always originally came with them and they are all the same. The early remotes that are harder to find are the wired ones that RCA and a few other companies offered as an extra cost accessory like for an 811K, but the Philco Mystery control sets always included the remote from the factory.
Regards
Arran