Not exactly what happened to break things, but for some reason, apache2 decided to be stupid. Every five minutes, virtualmin checks to make sure services are running properly and does some resets of certain permissions if it finds them broken/wrong.
Alas, while it had been working fine for quite a long time, suddenly it was re-setting them to something note working, breaking sites every five minutes.
It took a lot of troubleshooting to go through other things that were breaking things and finally finding the cron job every five minutes. During that time, a number of reboots were done along with every five minutes of sites going down (which I didn't realize for a while for various reasons).
At this point, the server has survived two every-five-minute cron jobs, so I feel more confident this issue is resolved, at least for now.
I still don't know exactly what changed. I did install the podman plugin - but things worked for a whole day after. I did do some system updates, and that might be a proximate cause.
Who the heck knows. I'm unhappy not knowing a root cause, but I will have to be happy with things working again. For now.