And if some user X posted something about a dispute they were having with chess tournament organizer Y, and Y felt this was untrue and decided to sue X for libel, I'd also have responsibility for that alleged libel because I'm the publisher?
Before anyone says that this is a ridiculous scenario, that's pretty similar to that happened with Prodigy's bulletin boards before the CDA. Prodigy had content guidelines for their boards, moderators that moderated to enforce those guidelines, and software that screened for offensive language.
Someone posted a message that someone else felt was libelous, and that someone else sued both the poster and Prodigy. The court ruled that because Prodigy enforced content guidelines and had filters for bad language, they were a publisher and responsible for the content of all the messages.
It was that case that was one of the main inspirations for the CDA.
Before that, the way the case law was shaping up there were only going to be two options for an online forum.
1. If the forum does not want to be liable for what its users say, it cannot moderate or restrict their content, except as required by law.
2. If the forum places any restrictions on content, then it is essentially in the same position as a newspaper or a book publisher, and to keep its legal risks under control it is probably going to have to do a similar level of fact checking.