Correct, your account only holds the playlists/songs that you have added. Songs and Playlists downloaded are independent to the device.
For example, I have 3 devices using the same Apple Account. They all have the same playlists synced because they are using the same account and the same songs in my Library, but each device can have different songs downloaded for offline use. As stated already, that is consistent with the other streamers as well. Spotify is the same way with downloading either the song or playlist for offline use.
https://support.spotify.com/us/article/listen-offline/
I don't know why if you had 5,000 songs in your library, you would want an option to download every song for offline use to take up storage space on your device, but that is what feedback is for and maybe others share your same desire. The advantage to a streaming service for me is to be able to play the songs without them taking up storage space. If I am going to be on a plane without a data connection, then I will choose a couple of playlists to download, I certainly would not need all 5,000 songs.