This sounds like an exotic problem, and a challenge to any technician.
But, I am going to take a guess, that you have a pressure switch
problem. When the pressure switch is satisfied after the water fills
to a correct level, the internal switch sends power out to the motor or
timer for agitation. If that internal switch is broken, it won't
agitate. But, when you put your timer to that sweet spot on the dial
and getting it to agitate, you are actually bypassing the fill switch,
because that spot may be a technician's test point on the dial.
To test my theory, fill the machine until it stops, then test for power
in and power out on the pressure switch. You will need to ID the
corrct wires by looking at the wiring diagram. One is L1, one is power
to the fill valve, and the 3rd wire is the one you want to test for.
This theory explains why it won't agitate through the first part of the
cycle, but it doesn't explain why it doesn't advance through spin.
Typically, a no-advance problem is almost always timer related.