Thanks for the log files.
The last log entry is dated July 19, and it was running PMS 1.42.0.9975.
It is possible the permissions for the Plex Data Folder are incorrect after you restored the files. That can easily happen on Synology NAS.
You should also install the current public release, PMS v1.42.2.10156.
Suggest you take the following steps:
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Follow FAQ 32: Repairing the PlexMediaServer shared folder. This ensures permissions are correctly set for the PlexMediaServer shared folder.
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Do not start Plex Media Server after repairing permissions.
You have 1.41.5 installed. Running it will perform unnecessary database migration tasks. It probably will not hurt anything, but there is no need to risk it. -
Download PMS 1.42.2.10156 from plex.tv/downloads and manually install it using Package Center.
You want the DSM 7, Intel/AMD 64-bit download (https://downloads.plex.tv/plex-media-server-new/1.42.2.10156-f737b826c/synology-dsm7/PlexMediaServer-1.42.2.10156-f737b826c-x86_64_DSM7.spk). -
Login to Plex Media Server
From a system on the same network as the NAS, point a browser at http://192.168.0.253:32400/web (use https if needed). You should be able to login and reach server settings. Make sure the server is claimed (Settings → server_name → General).
That should get you up and running. If it does not, report back with info on where things fail.
