PSA is public service announcement, if anyone was wondering.
I was having many problems with my MacBook Air M1, including Mail not remembering its columns, some applications failing to close when I quit them, Time Machine failing to find its backup disks, the Search field in System Preferences going gray, the display going black for a few seconds, and I’m sure there were other issues that I don’t recall. For no reason that I could rationalize, I deleted the Cisco AnyConnect (VPN software) folder from Applications, I deleted the three Cisco Services from the Network pane of System Preferences, and I restarted for the eleventeenth time. It has only been a day, but none of my problems have appeared since.
If anyone is having problems and running Cisco AnyConnect, you might want to delete it.
And if anyone has a recommendation for a VPN service or client (including Cisco, if you think my installation was bad), please comment. And if anyone wants more gory details about the misbehavior of the MBA, just ask.