Well, no guarantees but here is what I think is going on.

It is likely your model has a neutral drain, which means the timer must pause after the drain in order for it to go into the spin mode. It appears the timer is not doing this.

When you manually move the timer you are creating the necessary pause, so it spins.

You probably need to replace the timer, here is the part

Part Details - WHIRLPOOL Timer, part number: 661649

but if you decide not to take the risk that this will fix it, or don't want to put this much money into the machine you can just manually pause the machine and then re-engage it to make it spin. Somewhat of a bit of trouble, but a risk free, cost free solution.

See we like to give you as inexpensive a solution as possible!

Good luck.