The motor must pause before it can reverse direction for drain/spin. You are pausing it by lifting the lid or unplugging it. The circuit board, which has no relay, in this unit is for temperature control, has nothing to do with motor control.
This is a model with motor/inverter and mode shifter but a failed mode shifter coil would not cause the washer to stop operating. It would just cause the basket to agitate in agitation mode. May be a bad mode shifter and motor/inverter. Mode shifter should read about 98 ohms or 65 ohms, ,depending on version.
Pressure switch is closed with tub empty. When full, it opens. Pulling one wire connector off the switch while it's filling is easy and rules out anything to do with the switch including hose problems. A clicking relay doesn't mean the relay is bad, could be the control signal to the relay from the microprocessor. EEPROM is the memory chip with the program algorithm that runs the washer. The microprocessor does a "checksum" test on the data stored in it and if not correct (corrupt data), it gives error code.
You got that right. Don't even bother with the replacement Whirlpool design. The only decent toploader now is the Speed Queen ($800) and even it has dumbed down water level and temp control. IMO, your better off fixing what you have or looking for another good used direct drive.
Washer doesn't have a mechanical timer. As the washer is filling, unplug the connector at the water pressure switch. If it does not stop filling, unplug the washer. If it stops at that point, you likely have a faulty control board. If it doesn't, the water inlet valve is faulty. Did you test agitate function in service mode? Both high and low and speed didn't change?
The top part of the agitator (auger) is only driven in the clockwise direction when the lower agitator goes clockwise. When the lower agitator goes counterclockwise, the auger is not driven (free floating).
Ok, need higher resolution photo of wiring diagram, hard to read the wire colors and timer terminal numbers but looks like:
Cut gray jumper between terminal 15 and 17 at timer connector. Connect gray wire from terminal 17 to blu/blk wire that runs from timer terminal 13 to L terminal on temp switch. Also cut gray wire from terminal 15 at timer connector and blu/pink wire from terminal 16 at timer and splice those wires together. Now rinse temp will be same as wash temp. In photo below, red lines show wire cuts, blue shows new connections.