Earlier quoted context omitted.
It can be a private company it just needs to be regulated. I think you are underestimating how big Twitter is... It's way bigger than CNN its way bigger than Fox in fact those networks regularly quote tweets.
By eyeball-hours, Fox is much bigger than Twitter among American audiences. Viewers tune into it for hours, on a daily basis. What kind of regulation is it subjected to? It doesn't have to be neutral, and it has a legally-tested right to lie, and mislead. And that's for first-party content directly produced by it.
You can read the order. Section 230 of the Communication Decency Act set out provisions protecting internet service providers from liability for obscene, lewd, etc content on the basis they are a neutral platform (like an email provider) and not an active arbiter and curator of content published on their page. The law clearly states it's protection for private blocking and screening of offensive material (lewd, lascivious, filthy, excessive violent, harassing, etc). Not for editorial protection. Not for arbitrary enforcement of TOS.