But only for the same reason, so that they wouldn't be sued for hosting copyrighted material. The DMCA simply gives an official way to not have liability. The problem of overzealous litigation would still be there without the DMCA, unless we tackle that, then content providers are going to be quick to take things down so as to not be liable for hosting the content in the first place.

What I'm saying is that no, they won't do it to not harm their business relationship, they'll do it because otherwise, they'd be potentially sued for hosting copyrighted material without authorization.

If copyright rules were more lax, I'd doubt they'd be willing to cede nearly as much ground to NBC Universal and the like with Content ID and Content Verification Program.