PMS - Doesn't Start Anymore, Tray Icon Disappears Immediately

Did you clean out the Plex registry branch?
If not, it might contain a directive to use a specific folder as the plex data folder. If that folder location is not existent, Plex server will close immediately on startup.
i.e. if you ever this this: [HowTo] An extended guide on how to move the Plex data folder on Windows