but since some people seem to believe that the First Amendment protection of speech somehow should be imposed on private companies' Internet-facing content
That's not the argument. No one is saying companies should be forced to comply with the first amendment. "Free speech" as a principle exists outside of the Constitution...it's the shared cultural value that the Constitution seeks to protect. The cultural value came first, then the legal protection. This is how descriptive legal systems like english common law came to be.
The point is that these online platforms like Twitter don't hold this cultural value. That's okay, I guess, you don't have to. But they claim to, while saying they're only denying a platform for "hate speech," but this is bullshit too since they never punish anyone for saying "killing white people." That's pretty damn hateful.
They're not for free speech, they're not against "hate" on principle, they're just partisans.