This is an interpretation that is not so black & white in reality and probably depends on jurisdiction to a great extent. In the US, the courts disagree with you. Your statement resembles how things worked prior to the CDA in 1996.

Now, even moderated outlets have safe harbor protections under the DMCA and similar protections under Section 230 and Section 509 of the CDA which determined that internet services were not (automatically) publishers. Courts have held that outlets are in the clear if they make a reasonable effort to identify illegal content and also take action when something is brought to their attention.