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Notice: Something Better Exists
After working on this for a few weeks, I found dufs, which I am now using instead of this.
webfs
It's a WebDAV server that generates and caches media thumbnails.
I front-end it with Caddy, which handles authentication and access controls. Because the thumbnails live at the same path as the original media, so I don't need any special gubbins to protect thumbnails the same way.
Thumbnails
If you put ?thumbnail at the end of the URL,
it will use ffmpeg to generate a WebP thumbnail that fits in a 320x200 box.
I use WebP because it provides alpha channels, animations, and good compression. If your browser doesn't have WebP support, that's a real pity, and this isn't a good software solution for you.
Thumbnails are cached. No attempt is made to verify whether thumbnails are up-to-date, or to clean up the thumbnail directory. Something else is going to have to manage that.