Regardless of the legality of the action or the product itself, a video reviewing, showing or reporting on it shouldn’t be passable of a copyright claim.
Even if the video shows copyrighted material, it still shouldn’t be allowed for Nintendo to claim it, as that would fall under fair use. Just showing a few screenshots of a video game for the purposes of education in an otherwise unrelated video would never fall under copyright infringement.
The piracy argument has nothing to do with Nintendo claiming a video as their own, despite them having no rights to do so.
The “theorem”, if we wanna call it that, says that, given an infinite amount of monkeys and time, they could write Shakespeare.
This doesn’t mean it’s actually possible in the real world, it’s just to say that random events can seem, from the outside, like intelligent creations. Like a cloud that looks like a pig, no one actually created it to look like that, it was just random happenstance.