I don't think that's true, the law generally doesn't apply if you unknowingly are in possession of Illegal goods. So it would need to be proven that the host knew it was illegal.

Do you have evidence that a host was successfully held legally liable for unknowingly hosting illegal/unlicensed content? The Napster case is the closest I know of (post DMCA though), and that centered on whether they knew about the infringing content, as well as whether they promptly removed infringing content once they knew about it. This is certainly related to safe harbor provisions, but I'm interested to know of any examples before the DMCA was in place, because I have my doubts that they're truly necessary. To me it sounds like a way to hand even more control to copyright holders since they merely need to send a notice-and-takedown request instead of actually proving anything (at least that's my reading of Section 512 of Title 17).

That said, they're are better ways to handle it imo. For example, strengthening rights and responsibilities of individuals over submitted content could force hosts to explicitly partner with them to profit from it, which means they'll share liability and thus be expected to put more effort into verifying legality.