I was able to solve the problem!
I had the exact same issue. I couldn't even play songs that weren't already downloaded.
New songs from the online Apple Music library started loading and immediately cancelled without playing.
Apple support is informed and they investigate in the issue.
However I found a solution on my own (it is not a VPN problem like Apple support suggests).
The cause of the problem seems to be when you previously used iTunes.
Apple Music goes through all songs that you've dragged into iTunes and tries to find the corresponding song in the online Apple Music library. This process breaks the entire Apple Music app in my case (even after it has completed).
I resolved the issue by:
- Deleting my entire media library in Apple Music.
- Reinstalling Apple Music again (important, otherwise it won't work).
No I can play any song, download them, etc. without any problem.