Twitch and Facebook and other sites that allow you to stream video just need to get with the program and do what YouTube has done. If YouTube detects content that is copyrighted (music) it simply monetizes the video and pays the copyright holders. The problem is solved. No strikes, no copyright issues etc etc. The downside is content creators cant make money from streaming content with music in it.
Solution #2 is just kill the audio track from the machine. That solves the problem as well but takes away part of the experience. Unless the copyright holders are going to argue that simply showing a photo or video of a copyrighted image on your stream is a no go. If so good luck posting videos of your trip to Disneyland etc.