The way it works now is that the accuser has a few days to submit proof that they're suing to defend their rights, or else the platform can reinstate the content. In practice this would (and does) lead to a game of whack-a-mole between large rightsholders who have to pay money for lawyers and uploaders, who don't pay anything but an internet bill. Obviously, this doesn't fly in court and platforms have to go out of their way to ensure that they aren't profiting from mass piracy. It doesn't help that the aforementioned lawyers are always eager to go after a juicy, solvent target instead of Some Dude in Ohio with fiber and a lot of free time.

What do you do when your basic fair use check turns out to not be so basic after all? What happens if a video starts as academic but later turns out to be part of a commercial operation? Is the platform indemnified because it was "obvious?"

You're also forgetting that the platforms do not want to take down content. YouTube at least does a few basic checks automatically and makes heavy use of human reviews. I'm sure a few people would benefit if they quadrupled their spending on copyright review, but it's crazy to think that it makes sense for them to do this.