Kenmore 110-28002011 top loading washing machine.

This model wants to lock the lid and give the basin a little spin (measure the stuff inside I assume) before actually adding water and running. Normally, this is:
1) Press the start button
2) Listen to CLICK as the lid locks
3) Listen to motor startup as the basin spins
4) Listen to basin slowing down on its own
5) Listen to water filling basin.

As of a few days ago, this broke. Now I get this:
1) Press the start button
2) Listen to CLICK as the lid locks
3) Listen to very quiet hum like an old mechanical timer

I did try the chat function "Chat with a Pro" yesterday, but I had to repeat what I already typed into the chat window which didn't fill me with confidence. Also, my browser crashed, so need help finding the part again anyway.

The agent told me the part I needed was a logic board or circuit board or something. If that was the problem, I wouldn't have expected the quiet hum. Does that make sense? If the problem is the $250 part, do you recommend fixing an 8 year old appliance? Or am I likely just around the corner from the next $250 part?

Thanks for the help.