Try creating a new shared folder for your media, not using Public or any of the other Synology default shared folders.
The Installation & Setup of Plex Media Server on Synology and Synology FAQ (DSM 5 & 6) both state to not use the pre-defined shared folders.
Neither document specifically mentions Public. However, it is on the list of Synology pre-defined folders (linked above).
As a test, make a new shared folder, such as /volume1/Media/, add a Movies subdirectory, copy over a few files, check permissions, then create a new movie library and point it at the new location.
FYI, both the Install & Setup and FAQ are very good documents. Give them a read if you have not done so. They definitely help when running Plex on a Synology NAS.
