You're missing the point: If they have to vet every single piece of information, that's not an internet connection at all; they won't be able to allow any message boards, any third-party hosting, any user-created content at all. Because without Common Carrier status, they're legally liable for any and all of it that is transmitted across their network. Someone emails someone else about a drug deal? They're guilty of co-conspiracy. Post a death threat? Company's now a co-defendant for delivering that threat. Someone pirates movies/software? They're liable for pirating, too (they "copied" it for you).
Without common carrier protection, any ISP will effectively have to disconnect themselves from the internet to avoid being buried under a mountain of lawsuits ... which may happen, with a conglomerate of sites hosting their own content and only talking to each other, not allowing email or instant messaging .... and then everyone will drop them and go to other ISPs that are, y'know, actually connected to the internet. Sorta like what happened to AOL/Compuserve/etc., but much faster, because the wider internet is already known to the populous.