For music at least, you can get the best of both worlds with Google Music. Any songs you purchase, you own. You can download them, back them up, migrate them to a different service, whatever. For songs you don't own, you can still stream them and save them offline as long as you're subscribed.

AFAIK no real equivalent to this exists in the video streaming world yet. Plex is starting to come close though: https://www.engadget.com/2016/09/26/plex-cloud-online-server...