I spent much of the last day fiddling with a Whirlpool.
Turns out is was a faulty Wig-Wag.
I kid you not. It looks like;
except in the picture above it is turned upside down from how the device is oriented in the washing machine.
...thirty bucks or so...and the thing should be good for a few more years I think.
Symptom. Machine fills, agitates, drains, but does not "spin".
One of the two barrel shaped (black colored in the photo) solenoids gave up the ghost. Crossing a couple of wires, then testing it in "spin" cycle made the agitator function.
That indicates the non-functioning side was indeed being energized, and isolates from "timer" malfunction and potential lid switch problems too. Of course I had already gone to the trouble of bypassing the lid switch -- but given all the other conditions, the test results did isolate the problem to the one solenoid, with that being confirmed by lack of continuity using a Volt/Ohm meter, while the other solenoid (the agitator side -- which was still functioning) gave enough resistance (ohms) but not too low as in shorted, while the "spin" solenoid having none at all -- showed a break.