The mystery is solved! Bill spent some zen time with the dryer circuit diagram and figured that the problem had to be in one of two places. I'll pass the discussion over to him now.
The picture below is the disassembled centrifugal switch on the dryer's induction motor.
Induction motors can't get started unless they have a separate start winding, or a shade pole (don't worry if you don't understand this), or a capacitor in parallel with the winding that gets switched out, or some means of getting a rotating field to set them off running from a stopped position. But once the motor gets going, you have to switch out the start winding or the capacitor. This task is accomplished by the switch shown in the photo above. Incidentally, it also switches in the heating element once the dryer is spinning. This is to prevent the heating element from coming on if the motor is not running and the air flowing through the dryer.
The problem turned out to be that the switch was not closed on the start winding when the motor was off, which is has to be for the thing to get going. My friend Elmer and I thought we had fixed this in October, but we made a fatal mistake when we tested the motor on the bench and we powered both windings from the supply, thus bypassing this switch which needed to be closed. Ooops. So we didn't notice that the switch failed to make a connection. I have pondered the problem for a while, and even drew out the circuit diagram on the computer so I could make multiple copies, trace it, document the sequence of operations, and figure out what was wrong.
Last week I decided to sit with the diagram and my test results thus far finally isolate the problem. That was Thursday and I had the Zen moment not long after starting. The fault had to be in one of two places (and mercifully neither of them was on the control board for the reversing relay), and I had evidence to believe it was this switch, not the reversing relay. Today I performed the tests that confirmed it, then tore the thing apart and adjusted the contacts until it worked as needed. I let Jackie turn the start switch for the test run. It worked.
I did tax her patience (for which I apologize) for a few weeks by insisting that I wouldn't order parts until I knew what needed to be replaced (which in the end turned out to be nothing). She was seriously irritated about having to push start the dryer every 5 minutes, but I tried to convince her that our pioneering ancestors still would have seen this as a labour saving device :-)
And we might have bought $250 worth of stuff for nothing, not to mention labour costs. Repair technicians know less about this than I do — they just remove and replace until it works.
But now the inner workings of the dryer are completely known to me. It will be easy to keep it running for years to come. Back to Jackie.
I have to say that the "push start" of the dryer every 5 minutes did indeed tax my patience (pioneering ancestors aside). Especially on those days when I had 3 or 4 loads of laundry to do.
But all of that aside, I am glad that Bill figured it out and fixed it.
And now it is time to put a small girl to bed. Good night.