The legal obligations that platforms be neutral in order to not be liable for content on their website is a complete and utter fabrication in the public’s mind, and has no basis in US law.
It's not a complete fabrication, though to be sure the present debate has been painstakingly engineered over several years by multiple political factions.
Section 230 of the 1996 Communications Decency Act, which immunized from tort liability "interactive computer service[s]", was passed in response to a 1995 NY state court case that found Prodigy liable for statements posted on a forum by one of its users. See https://en.wikipedia.org/wiki/Stratton_Oakmont,_Inc._v._Prod.... The test the judge employed in that case was the degree to which Prodigy exercised editorial control over user-posted content, and in that case the judge found Prodigy exercised such sufficient editorial control that the discretion it effectively exercised in failing to remove the statement made Prodigy liable.
If Section 230 was revoked then that test of editorial control would presumably become the law in many if not most U.S. states as, AFAIU, the test wasn't created out of whole cloth, but rooted in well-established precedent. Some states might go another way but I doubt it; categorical, bright line limitations on liability are an unusual feature of judge-made law (which emphasizes fairness in the context of the particular parties, with less weight given to hypotheticals about society-wide impacts) and are typically created by statute. Other jurisdictions seem to have ended up applying very similar rules as the NY court, and even supposed Section 230 analogs (e.g. EU Directive 2000/31/EC) seem more like the NY rule in practical effect than Section 230's strong, categorical protections. Manifest editorial control seems like a sensible test for deciding when a failure to remove constitutes negligence; sensible, at least, if you're going to depart from strong Section 230-like protections. But I would expect significant variance in the degree of control required to be exhibited absent a national rule. In any event, massive sites like Twitter and Facebook might be faced with some stark choices--go all-in on censorship, or take a completely hands-off approach a la Usenet.
You're correct that there is some basis for it in past precedent, which was corrected in Section 230. But given the law as it stands today, the idea is completely false (even if it's not completely made up).
