Thanks for the many posts! Some quick responses and an update.
Re Japan: Yes, similar to Sydney Australia when I was growing up, though I understand its changed a little now with new construction. In the one or two cold months of the year, we'd all gather in the family room with the oil/electric heater and then head off to bed with a hot water bottle.
Re AR: new gas bill after new heater: Wow, that's really disappointing. Does the house feel generally warmer? The one thing that intrigued me about a new heater was the multi-speed options. Instead of "30 mins of heat followed by 60 mins of gradually cooling, repeat", I imagined that having half heat for much of that time would be comfortable in ways like: less hot/cold pockets in the house, less noise from the fan running full speed. Do you have pg&e with smart metering? You might be able to go online and compare before and after in the daily history they keep and get a better idea of what's going on.
Re new relay: I thought about that option. I decided against because: there are 2 relays and I wasn't sure which was which, there were a few capacitors on the board (perhaps used for timing?) and I was concerned that after almost 30 years they might be dried out. Easy enough to replace them too, but I find that sort of thing tends to snowball. Plus I remember pricing relays once before and they weren't cheap!
Re Skyn_Flint: Hope you're not freezing there!
So, on to the current status: In brief, I replaced the board about 5 days ago and its been working great (touch wood)!
Longer version: I found the board being resold at Amazon by multiple vendors and picked the cheapest for $55 and placed the order and waited. About a week later I got a phone call apologizing that the board was out of stock and did I want to cancel or wait. Doh. Eager to get it done, I found it locally for pickup at $85 + tax ... and went back to looking online. Found it again for $65 with shipping & no tax and ordered it.
The board arrived a week later and I was quite impressed. They re-engineered the original board beautifully for this purpose. Every connector was the same spade connector, and in the same place (except two were swapped) and well marked. The relays were replaced by (seemingly) solid state little black boxes, and most of the discrete circuitry was replaced by a single larger IC of some sort. Physically, its an identical size and the screw holes etc. all line up.
I restrained myself from just "going for it" and took the time to take a couple of pics of the existing board in place, drew a pencil diagram of all the wires, and then proceeded to pull them off one board and put them on the other, one by one. About 1/2 way through, I ran out of room for both boards and had to pull all the remaining connectors and remove the old board completely. The rest of the wires plugged in without problems. The screw connectors for the thermostat were slightly different (a terminal block vs individual screws on the board), but the order was the same and it was obvious.
The only tools I needed were a small hex nut driver (part of the cheap socket set I bought long ago for some other job), a screwdriver, and needle nose pliers. I used the hex nut screwdriver to remove some venting to give better access to the control box, and also the cover for the board. The pliers helped remove the automotive style spade connectors and the screwdriver for the 2 mounting screws and the thermostat wires. The whole job took ~45 mins, including the time to document beforehand.
So far, so good. The heater runs exactly the same, except without the fan stopping/starting 1-3 times at the end of each run.
Still to come: removing and balancing the blower. I was tempted to do it the same day, but decided it was best to reduce the number of variables and make sure that the new board was doing its job before throwing in messing around with the motor. I'll attempt the motor later this week. The hardest part there is removing more venting so I can get access to it and pull it out. Also, it looks like there is intentional clearance around the control board box so that I can leave it in place, but I won't know for sure until I try.
I'll keep you posted...
And I'm still royally pissed at the HVAC people who suggested $500 parts + labour to do this same job while all the while saying "this heater is so old, you'll just change out one part and another will fail". I reckon its just the opposite (sorry AR, but your experience was a crucial part of the nudge to try it myself). Put in a new heater with all its complexities and watch parts fail left right and center in about 10 years (as soon as its out of warranty). These older heaters are so simple, it can likely be repaired for many years to come. Especially when there are up-to-date replacements like the new control board available (and presumably similar advances with new blower motors, gas sensors etc).
Thanks again everyone for your encouragement and insights!