I had this issue and for me it was plex media server not detecting that I was on the same subnet (even though I was) and hence not activating the server setup process. Instead, like the op, the initial signing in at the plex media local lan ip acted like a client with no media. To get it to work I had to reinstall in host mode, which sadly meant tailscale being unavailable at container level.
EDIT: Okay, bridge mode does work. After claiming the server while in host mode and adding the lan details under the "LAN NETWORKS" setting within the plex settings, I rebuilt the container under bridge mode with tailscale enabled and everything is happy.
One quirk I found was that I had to turn off advertising the lan ip of unraid server as a subnet route for tailscale running on the unraid server. Otherwise machines connected to the tailscale network and the lan would still appear as remote with their ip addresses being the docker bridge network address (172.17.x.x).