ok, so no snap. a quick internet search of "/run/media/<user>" seems to point to a new automount location maybe. my ubuntu machine doesn't do this yet. That would probably happen when you log in, so if you want plex to run without you logging in, you would need to manually add it to fstab and put it somewhere else.
On the permissions issue, it is possible that the user id of the plex user changed so it no longer has access (depends on who owned the directory structure in the last environment) Once you get the new location for the media drive, I would just chown or chmod everything that you want plex to have access to (but that's just me). Once you have it mounted, you could check the owner, group and permissions of a directory (not where the drive is mounted, use some subdirectory) with "ls -l", and we can look at those.
Sorry if I intruded on anyone, I've been following the list, but the plex question struck a chord with me, I had this problem with a snap install before (and not realizing I was using a snap).