This morning, I pulled the plug, and then, one by one, disconnected *every* wire I could, and cleaned the contact surfaces, and reconnected them.
I also blew compressed air over the boards, and tapped on each solenoid of the gas valve and relays in the fan-control PCB.
This took about an hour. When I powered it back up, I heard a click click click, and then a small whoosh, and then after a while, a bigger whoosh, and finally the blower.
The house is toasty right now - but it has only been working for a couple of hours. Thanks for all the help & advice!
I'll report back, to let you know how it progresses along over the next day or three ...