Running version 2025.7.0 (888), my local Plex server constantly shows offline. All the remote servers share we me work fine, and the Preview app on iOS can also see my local server. It seems the Apple TV app is missing the Local Network permission, but I can’t find a way to trigger it or set it.
There’s no separate „local network access“ restriction on tvOS for apps.
Are you able to access your local server from the same Apple TV using the regular Plex app?
The latest build has a troubleshooting/debug menu in its settings with an option to handle logs – I haven’t yet looked into that option, so I don’t know how it works.