[Adam]

I replaced the triac and isolator, and it would turn on again with the switch - I did both at the same time, so I don't know which was bad.

That resistor was in-between the triac and the photoresistor side of the isolator - I indicate where it is in the schematic (which I put as the first image) - it looked like someone actually cut across the trace connecting those, then soldered the extra resistor in. But I tried connecting across it with a clip lead, and it didn't seem to make a difference either way, so I just left the resistor as it was.

But it still would not turn on/off with the remote. I tried peaking that coil, and then I got it so it would sometimes turn on only if I held the remote right up to the microphone and clicked it several times in a row. I was still getting a slightly higher voltage there (I indicate where I was checking it on the schematic) with the switch vs. the remote even after I adjusted the coil - but the problem turned out to be the leaky electrolytic capacitor I circled.