There are two things there. One is that notice and notice refines the responsibility appropriately.
No service provider wants bad guys on their network. The interesting thing is that so often there are innocent transgressions that notice and notice solves without the classic “record label taking grandma to court” story. Most people, when they hear about problems, simply go away.
It's the real offenders, the massive bad guys, who you are not going to get with notice and notice, or notice and takedown. There is where you need the big stick. When I'm talking about service providers helping, I think they really are the ones who, at the front lines, are doing a lot of that enforcement, because it's way easier to be smart about it up front and to have a straight game.