Plex asks me to "Get Plex Media Server" from within Plex Media Server

That part’s good.

I would like you to do the following, albeit it a bit extreme, to maximize chance of success in one step

  1. Go to the upper right corner of your Plex/Web UI
  2. Sign out of the app (dropdown is in your Avatar)
  3. Now open an incognito window
  4. Open http://192.168.1.101:32400/web

See if that brings you to a sign in and Setup Wizard start.
If not, we can dig further by manually grabbing the log files.

Go to the Syno FAQ (you’ll find some of these useful from time to time),

FAQ 1 - will grab the logs manually so I can inspect them.