Take the actual switch to an electronics tech. It may be able to be repaired, or cross referenced with something else. At this point, you have nothing to lose. I've taken switches apart that weren't meant to be and gotten them to work. Maybe even some spray contact cleaner may do the trick.
Nice idea, but it won't be a switch of the type we had when you and I were lads. The only switch will be a brief momentary switch that destabilizes another circuit, like an IC that contains some sort of flip/flop circuit. There are tons of variations on this theme. This is complex, but years ago everything was much lower powered. Even then switch contacts used to burn out switching AC circuits from the arc of the back EMF. With starting circuits of one sort or another, the momentary switch only momentarily makes/breaks a low voltage DC circuit. The switch that handles the power is solid state and contactless.
So that is why his unit switches off. The start button is flipping the circuit, but it is flipping back by itself after a short interval of time. It is very hard to fix this type of arrangement, without at least a circuit, to see how it is configured, so you can sort it out.
You are used to this I'm sure as the on/off button on your receiver will be this type of arrangement. There is no toggle switch is there?
