Background
I'm still working toward my goal of being able to leave my phone in a locker while I work out. Despite the gains of WatchOS 5, it appears that my only real option for music without my phone being nearby is Apple Music (the service) or moving audio files to the watch manually with Apple Music (the app). I'd prefer not to have Apple Music installed on my phone. Googling reveals a lot of articles from the Series 3 release saying that this should be possible now, but I've not found any actual solutions.
Actual Question
Can I listen to music from my Apple Watch without having my phone nearby and without requiring Apple Music be installed on my phone?
Best Answer
MixTape turns out to be an excellent solution that costs only $3.
http://dorshorst.com/mixtape/
I'm not sure what changed between now and Feb 5th when I first tried it (and commented above) when literally nothing worked but I tried it again today and it worked perfectly for my needs.
As pointed out by @dwightk, Pandora as of 2019 does not support streaming directly to the watch, but does allow you to manually transfer songs to the watch for offline, phone-free listening if you are a Premium subscriber.
So far those are the only two solutions I can find.