Good thing that wasn't the argument, then. In fact, your summary of it doesn't even make sense -- middlemen don't get in trouble for taking things down, they get in trouble for not taking things down.
What actually happened was that just before the Internet got big, two cases came down concerning different online services. CompuServe got sued for user-posted content, but was found not to be liable because they had not moderated anything and were just a middleman. Prodigy got sued for user-posted content and was found to be liable because they moderated their boards (for things like bad language; they wanted to be family friendly) but had failed to moderate every post perfectly. By letting one bad thing through, they were liable for it -- and by extension, anything else they had failed to catch.
Since Congress wanted sites to moderate user content -- they were really concerned about porn -- they passed a law that encouraged sites to do moderation but did not hold them responsible for failing to moderate every single little thing perfectly in every respect. Further, sites got to choose what they were moderating for -- could be porn, but could just as easily be off-topic posts, like talking about carrots when everyone else is talking about money.
In practice, sites don't like to moderate much -- it takes effort, it may lose engaged users, it costs money, it can't please everyone -- but they certainly can, and there's nothing wrong with it. Get rid of the protection of the CDA and sites won't be able to do mandatory moderation sufficiently, so they'll fall back on none. This is apparently okay with scum who get kicked off of boards left and right, but should not be okay with people who have standards and don't want to put up with that crap.