I'm pretty sure your edit nails it: flatpak on my moldy Debian doesn't work correctly as a non-root user, and the alternative of running it as root and then running the application as a normal user was neither intended nor foreseen.

I did finally install Debian testing, and it looks like the flatpak problems are resolved, but I decided not to bother installing foliate because of the nearly 700MB of dependencies it wanted over my rural satellite connection. C'est la vie.

I went ahead and burned the data: for Science! After the dependencies were all downloaded, it works correctly and completely. It's a fairly nice ebook reader application, and it only took three days and replacing my operating system to get it.