The Pruneyard decision [1] though did conclude that the sidewalks (even privately owned walkways) outside a business were quasi-public spaces analogous to public squares.
[1] https://www.law.cornell.edu/supremecourt/text/447/74
About your specific question, I would say "standing INSIDE the mall screaming" (screaming anything) would be more similar to posting large amounts of text on someone else's personal Twitter feed.
Whereas simply having your own personal Twitter and saying things that someone doesn't like would be more like standing on the walkway outside a business.
Also, I read the draft of Trump's order posted on HN last night, and it doesn't say that lewd or obscene content can't be removed. If you start using the N word or posting pornography, I think that's still legitimate for companies to remove.
The problem is that the idea of "hate speech" is being weaponized. If someone simply wears a MAGA hat, that's being called "threatening" or "racist". If someone expresses economically protectionist views, they're called "racist" even though protectionism was used centuries ago to protect European countries from other European countries and has nothing to do with "race".
The right could similarly weaponize the idea of "hate speech" by saying that any time anyone mentions any kind of social program spending, that's "threatening" because that's akin to "communism, which has killed hundreds of millions of people".
Now, plenty of conservatives do say they're concerned about the slippery slope, but they don't immediately equate "we should give housing to the homeless" with "put the rich in the gulags", the way the vocal far left equates "maybe we should reduce taxes a bit" with "they're racists who want poor minority people to starve to death".