You can stream Apple Music over WiFi/cellular without the phone. Not sure if that will include Spotify at some point but I do know the new watchOS has opened up a lot of those APIs.
Saving songs is spotty too, some of the "saved" songs on my phone don't actually have the data and won't play when I'm offline. I can't figure out how to force them to redownload short of maybe uninstalling the app, so sucks to be me I guess. Spotify's mobile app is so bad that I'm constantly reconsidering my subscription. I'd go so far as to say that being able to explore new artists and save their discography for later when I'm commuting is a meaningful improvement to my quality of life, but Spotify has managed to make the experience so shitty that I usually just turn on the radio instead.