I'd say you got a hold of another bad timer. There is really nothing else it can be. As you say, this is a very basic bottom line machine. Just a single speed motor, timer, water level and temp switch. The problem is most certainly not the motor. You do realize that gentle cycle uses an intermittent agitation? It only agitates for a couple of 15 second periods and just soaks for the rest of the wash in gentle cycle. When you say it won't cycle through normal or gentle, what exactly do you mean? Does it fill but won't agitate? What about drain or spin? Be more specific.

Whirlpool W10199989 Timer - AppliancePartsPros.com

Eric

It has a single speed motor so there's only one agitation speed. About the only difference between heavy duty and normal is the wash cycle timing. Up to 18 minutes for heavy duty and 10 minutes for normal. Just a marketing thing to make you think your getting more when your really not.

Eric

It's called delicate cycle on this washer, same thing as gentle. On this washer, delicate agitation is mostly just soak, then it agitates for 15 seconds and then soaks, agitates again for 15 seconds and soaks some more before drain/spin and rinse cycle. You can set the time for whatever you want on heavy or normal wash cycles, it's just that the max time for heavy is 18 minutes and for normal it's 10 minutes. You can only get one agitation speed out of a single speed motor. Most of these direct drive models have two speed motors and some have three speeds. Yours is single and it's high speed only.

Eric