Well, that's a single speed motor. Every reference I can find for this model number washer says it's a 2-speed motor. Anyhow, the part you need is here but I would try to get the cover off the switch, you might get lucky and just be able to clean the contacts for the start winding connections between the red wire from the wiring harness to the black wire in the motor.

Eric

Unfortunately, when I put everything back together, we had the same problem, I can start it by hand, but it won't start on it's own. So I took it apart again, inspected the switch, and everything looks good, clean contacts, etc. no burn marks. It was stiff to turn (I don't recall if it was more or less stiff than before), is there something that needs more lubing or something? As I say, it works just fine once I hand crank the motor.

More thoughts?

How do I double check the motor windings?
I checked the switch after this past time, having already cleaned it, then using it again, I feared the same blackening, but no, the switch was clean.
I had to remove the switch from the motor, there's only 1 screw holding the whole thing together, i.e. the cover on, but also holding the switch body to the motor.
I tested the motor alone after cleaning the switch, and it worked fine, it kicked in immediately. I put the whole washer back together and got only humming, so took it apart again and saw the same problem, it would start if I cranked it by hand, but not on it's own.
I can take pictures or even short videos of anything you want if you think that will help. All of the motor connectors to the switch are in good shape, clean and quite tight to get on, so there's a good connection strength there. Could it be a mechanical thing, that there's too much strain on the motor from something?

Can I bypass that whole red wire by connecting the black to the capacitor terminal itself?

Absolutely nothing. No power surge, no hum, no buzz, no movement, nothing.