Plex is built as a ‘server’. Every communication it does uses networking infrastructure.
But there could be an issue with your router which may cause this behaviour.
To verify this, do the following:
Settings - Server - Network - ‘Secure Connections’ = “Disabled”
afterwards, restart both Plex server and your web browser.
Then go to Settings - Web - ‘Allow Fallback to Insecure Connections’ = “Always”
then reload the ‘web page’ with F5
Then repeat the playback test and keep an eye on your network monitor.
How does it compare?