How? Reaching contracts with all major copyright owners in order to stream your out-of-garage hobby?

Buying music (I assume that is what you meant), will not let streamers use it, thus there is no meaningful solution as of yet.

Just because contracts are difficult to get doesn't mean you have free rein to do whatever you want. There are plenty of meaningful solutions to this "problem". Don't play music at all. Play music you are allowed to play (be it paid or just music that allows streamers to use it). It may not be what people want to hear, but obviously the music is of value to shanelja's audience as they requesting songs.

That might be a solution, but as OP was saying, that is likely do draw people away.

Get a license with something like ASCAP[1]. I think that's what most businesses using music in a public setting do.

Legally using copyrighted music is not exactly an impossible task. I mean, it's basically the entire business of some radio stations, and some stations are quite small.

Now, is it worth the effort and cost for a Twitch stream? It depends on how much value the music brings. There's also CC or royalty-free music, if a streamer just wants some music playing.

[1] http://www.ascap.com/licensing/licensefinder

That does not give you license to steal other people's content and profit from it.

If you want music but you don't want to purchase rights, there is plenty of creative commons licensed music available.

Why is it that pointing out a problem requires one to provide a solution to that problem or the point is somehow invalid?

Learn to play the harp and compose your own songs.