Simply put Plex, as a platform, isn’t designed to function without Internet access. Clients will cache auth for a period of time but they expect to be able to contact their auth servers from time to time to refresh it.
There are certain things you can do to (partially) mitigate this. Specifically, you can configure:
Settings → [Select Server Name] → Network (Show Advanced) → List of IP addresses and networks that are allowed without auth
This will allow clients on the specified networks to access the server without authentication. But the client apps must support this, via manual connections.
There’s a feature suggestion to allow for local auth on PMS itself, but I don’t know how popular it is.
Here’s an older “how-to” which describes some mitigation steps you can take before an outage occurs but, like I said, it only applies to certain modern clients.
I use another media server as a backup. It’s not ready for primetime as its clients are, frankly, terrible. But any port in a storm.