A couple of times the washer wouldn't spin after final rinse, and I could open it, back up the timer, and it would go ahead and spin afterwards. Now it won't do it that way either. Timer seems to advance OK to the end, and I can hear something moving (motor I assume), but the tub won't spin.

Any advice on troubleshooting? Thanks!

Thanks for taking a look, model number is:

LSQ8243HQ0

Yes your lid switch has split apart
after years of closing the lid switch mechanism begins to separate.
Try accessing the lid switch and see if it seems to be split open. If so remove it then find some good strong zip ties and run 2 around it good and tight and replace it into the machine. I have repaired many this way and they are all still working fine.
In some cases it may not work but since you can push it with pencil I suspect this fix will work.

The split may not be apparent unless you depress the switch, at which time the case will separate thus not allowing it to activate. This will worsen over time. If your issue reoccurs, and I believe it will, take a couple of 8 to 10" zip ties and run each around the body of the switch, pulling them as tight as you can. I am adding a couple of pictures how-to. I could only find one zip tie so I show it in both positions.