Except that SCOTUS has already at least flirted with the idea (see Packingham v. North Carolina, 2017). Also the executive order in question doesn't have anything to do with public spaces but rather Section 230 protections; the argument is that fact checking is a form of editorializing.
I think it is pretty clear that fact checking is a form of editorializing. It seems that Section 230 was being applied even if the intermediary was editorializing the content.
It seems significant that Twitter wasn't a party to that case. People seem to be enthusiastically conflating it with some sort of ruling on what Twitter can or must do. I'm not sure if you are, but even bringing it up worries me.
Both Packingham and Knight involve government action to deny others access to social media. They don't involve any obligations of social media sites themselves to provide access.
