There are known issues with the Download function between the current PMS version 1.41.8.9831 and the Plex iOS app.
https://forums.plex.tv/t/unable-to-remotely-download-content-to-player-18-5-on-ios-when-not-on-my-local-network/921089/6
One current workaround is to downgrade to an earlier version:
If you’d like to try and rollback to an earlier version of PMS, here is the process that you would follow:
https://forums.plex.tv/t/synology-faq-questions-answers-and-how-tos/490215/39
Except on re-install you would chose Normal Installation, and use an install file of an earlier version.
All of your existing libraries and database entries would remain intact.
There is an archive of past plexmediaserver download urls on Github. Find the version you need and the link will download the version from the plex download server.
https://github.com/axlecrusher/plex_download_urls?tab=readme-ov-file
Be sure to choose one with the same exact OS as you are running on Synology.
If you are running DSM7.2, your server may already have saved earlier versions for re-install.
Some are having success with ver 1.41.7.9823
Remote download will likely not work, but Download on your local network should.