I have an idea for a workaround. If we can't play music on a Twitch stream, why not stream your music somewhere else? It's really a horrible solution, but if it's the only way you can share music with your stream...
The only one I know of is Turntable, but it's flashy and you don't have complete control over the music. I'm sure you could get a bot to regulate some things, but I don't think you can stop people from laming a song. If there's another way to do this live like Turntable but more like traditional streaming, that would be the best option. (Edit: Forgot that turntable is US only, so it's a horrible "solution")
To the people that are thinking it's a Twitch decision: it's not. It's not their fault that the RIAA is forcing Twitch to do this.
Edit: Of course, I mean this plus Virtual Audio Cable.