The thing is that just by streaming games or anything with a copyright attached to it, it's already technically illegal. All these game streams and the foundations of Twitch (and Youtube to a certain extent) exist in legal gray areas, but it's up to copyright holders to contest these things. Once google bought youtube they had to implement some sort of system immediately because there's now ample incentive for rights holders to sue youtube, because youtube is now google and google has a ton of money, and I imagine Twitch is in a similar situation.
Basically, because of how awful copyright laws are, content providers have to bend over backwards because huge swaths of content on the internet is technically illegal, and rights holders have a legal obligation to atleast attempt to enforce their copyright or else they become meaningless.
As awful as it is for Twitch, I have a feeling it's a shared fate for any sort of streaming site that gets large enough or wealthy enough. At some point they simply have too much to lose and once people know you have money they will sue you.