I made a few more attempts to revive the old Mac mini without success; trying to update beyond Yosemite seems to be futile. So I punted and bought a new Mac mini at Best Buy; unfortunately, the only configuration in stock there (or at the Apple Store) was rather small: 8GB memory with a 512GB SSD, which wouldn’t hold my entire Plex library, so I added an external 1TB SSD.
I spent most of the afternoon and evening installing software; now I’m copying my movies and music from the Time Machine backup to the external SSD. It will take all night; I’m not going to watch!