I don’t have a theory for what’s happening with your setup. It seems that the theory is that either your NAS or the Roon package itself can’t access the internet. I don’t understand this theory because other things would be breaking if your NAS can’t access the internet. Can you, for example, access the “Package” app, and if so, does it complain that it can’t access the internet?
Some of us that run our Roon cores on Synology and other NAS devices use Docker to do it. Docker containers are very efficient and Roon runs as well for me in Docker as it did as a package. Better, in fact, since the package was broken for a while when DSM 7 was released and also is, in my opinion, more complex due to the needs for package-specific accounts. The container-based approach just works.
I wrote a walkthrough in 2021 for how to set up a container in DSM 7. Now that DSM 7.2 supports compose files, I use that approach. But the DMS 7 approach still works even though you have to figure out the new UI equivalents of what I described in the walkthrough. Still…if you’re frustrated and want to give it a shot, this is probably enough to get you going: