I have an old (late-50s era) forced-air furnace system with a standing pilot. Over the past few years it's started randomly shutting off (an audible click occurs as though the thermocouple has cooled and tripped it). I've replaced the thermocouple and even the entire pilot/thermocouple assembly and had an HVAC-employed neighbor verify my work.
It doesn't coincide with windy days and wasn't a problem 4-5 years ago, so it's unlikely to be some kind of backdraft issue with the vent.
I'm now suspecting the (ancient) control system's transformer is on its last legs. My reasoning for this is because the Nest thermostat (1st generation) now and then reports it's lost power then randomly comes back on. This doesn't seem to coincide with the pilot shutoff and is much rarer, but I'm thinking because the Nest has got a very small power draw and I'm assuming holding open the safety valve is by far the larger draw, that's usually the first to trip.
Is this a reasonable theory? Should I look into replacing the control system?
[Removed alternate limit switch suspicions and pic.]