And Twitter enjoys no immunity here. If the President wants to sue them for defamation or libel for fact-checking them, he's welcome to. They own the fact-check copy. I'd have no issue with him suing them as a private citizen. Good luck with it, I say to him. Truth is an affirmative defense.
As for the rest... I don't know what to tell you. Freedom of the press does not imply requirement others listen, or obligation to provide presses to people. Demanding those things implies the ideas cannot stand on their own in the common marketplace.
(Baker is a different issue; sexual orientation is a protected class in law. Political views or "being President" is not, for self-evident reasons).
