Nothing is broadcasted to the internet.
The address app.plex.tv is only determining from where the Plex Web app is loaded into your browser.
Once this is done, it works the same as the local web app.
It needs to have internet connection, at least for the first few app strats, to
authenticate against your account
download any codecs it might need
I don’t know what you are trying to do there, since obviously you have internet connection in some way.
Be advised that you will run into these kind of problems again and again, if you are trying to use Plex Pass features, like Plex Home and Managed. Plex was not built to be used permanently offline.
Plex Web 4.x uses the new UNO interface. Some people still prefer the old version. The local version hasn’t been updated to allow users more time to use the old version until UNO becomes public to everyone. This update will probably occur within the next couple of PMS updates.