ISPs aren't common carriers and have never been common carriers. The FCC doesn't classify them as common carriers either. Really that's of no relevance though.
The DMCA allows that the ISP (if it is not the creator of the content) has a safe-harbor... which unfortunately has meant dick. See for example Viacom v. YouTube where Viacom sued Youtube and it took YEARS and MILLIONS OF DOLLARS for Youtube to prevail using that "safe harbor". That's not a safe harbor.
The DMCA is a piece of crap, written by idiots trying to compromise with an MPAA/RIAA hellbent on destroying rights to protect their "intellectual" property. The safe-harbor provisions including nothing for early termination of a lawsuit (like SLAPP laws) or fee shifting to make the loser pay the winner. Instead it's all about the "IP rights".
The MPAA and the RIAA and the BSA and... (MAFIAA for short) bastardized property law and trademark law and copyright law in an attempt to get the best of all worlds in one. As a result ALL OF US lose.
Please don't make your argument about child porn. It's like bringing up Nazis and ending the discussion. if you can't make your point without "think of the children" or "the Nazis win" you didn't have one to make to begin with.