Plex shows running but inaccessible

what’s the exact URL you use to open Plex locally?
same result for http vs. https?

what Synology model and DSM version are we talking bout?

and last not least… please manually collect the server logs. considering Plex isn’t running for you, you’ll need to check out the Plex data directory on the Synology and zip the Logs folder yourself:
/Volume1/Plex/Library/Application Support/Plex Media Server/Logs

The Plex data directory (/Volume1/Plex/) is hidden by default; you’ll need to grant your user permission to see it in the first place. When done, make sure to remove permissions again for the default user to avoid messing up the data directory