New sc-download setup can't access files even with full permissions

You are asking that in the wrong forum. The devs have nothing to do with the synocommunity. They kinda screwed the pooch for upgrades when they altered the user nzbdrone instead of just adding that user to the sc-download group like they did with deluge.

Sometime the best thing to do is to start “fresh”.

For instance, instead of trying to tweak folders packages and myself fiddling in the *nix layer have been messing with in the past, I just created a new shared folder for my download clients, gave sc-download permissions in the Syno layer, and stayed away from the *nix layer. Seems to work like a charm.

As far as restarting, updating, etc. just make a backup (see wiki for syno instructions), deinstall Sonarr package, reinstall the package, and restore the backup. That will mitigate any permission errors in the installation folders and databases.