The government is not forcing you to carry or not carry content, they're forcing you to take responsibility for what you selectively publish ("allow" as you put

The government is not forcing you to carry or not carry content, they're forcing you to take responsibility for what you selectively publish ("allow" as you put it) and not for content you simply repeat without looking at. It's the same principle as in other areas like when you know or should have known about a hazard on your property you become liable for it. In the case of a forum, moderation implies that you should have known what the content was and so are liable for it. If I was running a forum I'd certainly want a Chinese wall between hosting and moderation, and a record of all comments moderators actually viewed (from reports and through normal use) to use as an affirmative defense should 230 be interpreted as in this ruling.