Crazy idea: Can't a streamer just setup their own live streaming audio server (i.e. Shoutcast and Icecast) and tell users to listen to that while watching their Twitch? This way, their videos can get archived later for on-demand use and the livestream isn't affected /too/ much.
If you could get the time difference to within the duration of an average song, it wouldn't be so bad. It would be messed up if the streamer tries to sing along, but that's about it. I think the main reason viewers like streams with music is just to share in the streamer's tastes and perhaps discover new music.
Another reasons is that music makes it less awkward, most streamers aren't talking 100% of the time and there will be awkward silences. Also, for me at least, the addition of music makes it more of a casual viewing experience. I'll often have a twitch stream on my second monitor while working and it's not distracting at all.
Ah yup that makes sense, for some reason I was thinking the shoutcast stream would entirely handle the streamers audio, rather than just the music portion of the audio (which makes a lot more sense...)