Or more tactfully, if you're hosting it, we'll assume you're guilty because we don't want to deal with it, and you agreed to a very restrictive ToS that lets us take it down.

Google cannot know if you have the rights for any possible document. If you have a whole book it could be bought from Amazon or downloaded via Torrent.

There can even be the case where you have private conteacts that give you rights that Google cannot know.

I can inly thing that they will look for the low hanging fruit: a folder full of movies, music, and other content.

Not really, otherwise services that host user-generated content, like youtube, wouldn't exist.

What a hosting service needs to provide is a way for the user to flag content + provide an "acceptable" response time.

Another approach, a more proactive approach, is for the content owners to share the digital "finger print" of the IP (movies, music, etc.) with the hosting service, so that the hosting service can scan uploaded files and compare them.