ps showed that the PMS process had the env var PLEX_MEDIA_SERVER_APPLICATION_SUPPORT_DIR=/var/lib/plex so the error message is legit. That means either systemd isn’t doing what I want or PMS is changing its env var (seems rather unlikely).
Maybe it’s unique to Arch’s package for PMS, but the env vars aren’t set directly in the service file. Instead, there’s another file referenced from the service file using EnvironmentFile=/etc/conf.d/plexmediaserver and the vars are set in there. The wrinkle comes in that using a systemd drop-in to specify Environment= doesn’t override vars set in an EnvironmentFile (not the first time systemd has done something unexpected).
Solution: create a copy of /etc/conf.d/plexmediaserver (because it’s owned by the package) and change /etc/systemd/system/plexmediaserver.service.d/override.conf to say: