Nobody wants to worry about this stuff.
As long as the purpose of the stream is not just streaming music i don't see the problem however I'm certain the MPAA and RIAA would not agree.
Beside if you always play the usual 200 songs you listen to when are you going to find new songs to listen to.
Listening to music is a group activity a lot of the times and you find new songs by doing it.
There's only 3 ways to find new songs and 2 of them suck
1. Listening to top hits in hopes that a few match your taste
2. Using recommendation algorithms on Spotify and other streaming sites
3. And the on that give you the best results hanging out with like minded people in real life or on the internet and seeing what their listening to.
So the answer to the question "Why do you have to use copyrighted background music for streaming of a game" is "Because playing a game with the volume at zero kind of sucks".
edit: And obviously that's not the entirety of the concern; there are definitely some streamers who play essentially radio music (or anime music or whatever) over the soundtrack on broadcast. I agree that stuff like that doesn't necessarily need to happen. But using a broad brush to say "No broadcasting copyright music on twitchtube" is both not a good idea, and exactly what people are worried will happen.
why do you expect to be able to broadcast copyrighted music for your own enrichment without paying the copyright holder?
Buying music (I assume that is what you meant), will not let streamers use it, thus there is no meaningful solution as of yet.
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.
If you want music but you don't want to purchase rights, there is plenty of creative commons licensed music available.
I play an MMORPG called Runescape which has millions of players, I've been playing it since I was 12 and now I'm 21. I stream on Twitch once every few days and frequently have 1,000+ people watching me, subscribing, following and most importably putting eyeballs on Twitch's adverts (which is one of the largest concurrent none-featured community.)
If this happens I lose my anonymity (because Google+) which protects my account details from being stolen. If this happens I lose the ability to stream while playing music in the background, my followers even have the ability to request songs during the stream using an automated service, where does this leave me with regards to copyright then?
Even worse than this though is that if this happens, Google will in some way fundamentally change a service which I and my followers love to use - a service which they don't understand.
I speak for myself as a well known member of this community: If Google buys Twitch, Twitch will die.
This video basically (and very explosively) explains the current thinking of Twitch users: https://www.youtube.com/watch?v=loeFYdbvrZM (turn down your sound)