It happens occasionally that their servers go down and nobody is able to stream anything. I don't know why their servers are a dependancy in that chain at all, I probably looked it up before but I don't recall it being a good reason.
It's needed for authentication but locally, it should be nearly fully functional. At worst, you should need to set up a subnet that's allowed without auth in the server's network settings.
Aside from that, internet is mostly needed for downloading metadata and the server can work just fine without an internet connection.
Since I started using Plex near 10 years ago I’ve never experienced any sort of service unavailability.
You do need to be online though to set up a client for the first time to your local server, which is annoying, but has never been a problem for me.