This time, I decided to replace the ignition cable. It's a lot like replacing the spark plug wires in a car, but there's only one. I did make a bit of a change though, and used some 10-ga. wire. I added some tubing over the top of it to increase the insulation value.
The idea is that if the original ignition wire is corroded enough, it interferes with the flame sensor. The spark voltage can travel through it just fine, but the sensor voltage can't.
Replaced it without cleaning the ignitor, and it sparked right up.
Looking at the old cable, the spark plug connector was pretty gray with an oxidized film. I checked it with a DVM, and it certainly was having issues. I need to find a small stainless steel brush that I can pop on the Dremel tool and clean it off.
I doubt that this is a long-term fix. We'll see.