Good thought Graham.
I would also suggest checking the voltage required for the switch, if possible. I say this because on many current radio systems, the receiver side of things, are now designed to operate on from5 to 7.2 volts or more. My new BEC's are selectable to that range.
I have another thought. I have run into this situation, where a dedicated switch channel does not send a signal at the full range of signal width that is available on , say, proportional channels. If I recall correctly, a similar situation occurred with an electronic switch. I pulled many a hair, they weren't gray back then, and some how stumbled upon the answer when I plugged the switch into a proportional channel. Of course, that didn't work still, but when I added the trim button range, viola, the switch worked. I am not saying that this is the case here, but an option.
You may also have a bad relay, solder connection on the board, or many other possibilities.
With luck, your battery pack meeds charging. If you have a BEC that you caan power your receiver pack with, that may be worth a try as well.