No files will play on clients.

[TheDreadPirate]

Open a terminal and cd to the external drive mount like:

cd /to/path/of/mounted/drive/

Then execute:
sudo chmod -R 777 *

This command will give all permission to all files in the folder you execute it and all files and folders that is in below that path.

You might need to restart the Jellyfin server and scan the library (Please test only on one external drive).

I would recommand you to run Jellyfin in a docker, then you can set permissions for the folders in docker which is better.

This would work, but only if there isn't an ACL or some other mount parameter restricting the permissions.