This seems to have to do with the way the iTunes Server works - it essentially just enables your local iTunes to stream the content it has indexed. So your iTunes is not allowed to actually DO anything but listen to the songs.
Synology is not to blame, this is "the Apple way" to do it.
If you simply want to have your files located elsewhere (not on your hard drive) then you might think about using a network share as your iTunes library destination and not bother with the iTunes server at all. Then iTunes would behave "normally" and simply store all data on the NAS.
This is however only applicable if you do not plan to "use" (read: add music, create playlists, etc.) it on multiple macs (simultaneously).
Maybe your solution lies in a mixture of the two: Use the NAS as destination for your iTunes Library on one mac and use the iTunes Server to simply listen to the music from other devices (such as an iPhone or second mac). Or skip the iTunes server part completely and use the AudioStation App/Website.