iOS 10: Control Center for Music- Music Won't Play
The problem here is in YouTube itself. 

One way they try to get people to join YouTube Red is that you are only able to play YouTube videos in the background if you are a paid subscriber. You may have had a free trial that expired, or YouTube updated the way they check if you're active in the browser, but the only real way to fix this is a jailbreak or a YouTube red subscription.