That's what the law was before 230, except that it essentially applied to illegal content, as well, because of the company tried to actively moderate even for just illegal content then it became liable for all content (it had to be 100% right if it tried to moderate illegal content.)
Being liable for only known illegal content is distributor, not publisher, liability, which is what it superficially looks like 230 does on its face (courts have applied 230 to provide no liability, not even distributor liability; expressly imposing distributor-style liability would be a modest reform.)
Because the NYTimes pays staff to create it's content and exercises complete editorial control over said content, and thus is fully liable for the content published as news articles on its site. The NYTimes however is not liable for the content in the comments section on said news articles, though, they are allowed to remove comments that they do not wish to appear on their commenting platform.
Twitter is also liable for content that it publishes, eg: what the Twitter support account posts. But it is not liable for the content that I, dlp211, post to Twitter. Twitter still retains the right to moderate their platform as they see fit.
This applies to not just Twitter, but to every platform on the internet, from the Twitters and Facebooks and YouTubes to the MyNicheVBulletinBoard to Palor, 8Chan, and 4Chan.
Now I have mentioned previously that I am open to tweaks. While I haven't thought deeply about it, I would be open to considering a tweak along the lines that the platform becomes a publisher when it promotes content via human or computer decision making. This may or may not be a good idea after I think about it more deeply and discuss it with others, but the point is that I am not of the opinion that 230 is the end all be all but I am also of the opinion that I would rather live in a world with 230 than without it. I say that as someone that believes that their politics would be greatly benefitted by the repeal of 230.
IANL even if they called themselves a platform they would still be acting as a publisher and the law would treat them as such. For example they would be editing and curating content, paying for content from writers, and making the material available under their own name.