That lawsuit passed because it was both critical and a parody, so fell well into fair use.
Simply layering generic commentary over something isn't enough to pass the bar. If it were, MST3K wouldn't need clearance.
Parody, OTOH, is enough to pass the bar, which is why Weird Al does NOT need clearance for any of his work. It's also why porn movies and mockbusters can do their thing.
Most live streaming exists simply because the video game industry is younger and much more permissive (not to mention receptive to its audience) than the music or movie industries. But if it came down to a legal fight, someone who is streaming a heavily modified version of Mario 64, with different levels, and character skins (even a lewd one, where the goal is to sleep w/the Princess instead of saving her), would have a much stronger case ("I was parodying the family friendly nature of Nintendo") than someone who just played the original.
It would all depend on specifics of course, but the former would likely win if constructed well, while the latter would likely loose.