Not a precise analogy, but ...
re: "Social media websites have become our modern-day public square."
I am quite sure that someone once made the same argument about why one should be allowed to collect petition signatures (or hand out flyers, or ...) at shopping malls, and was shot down by arguments that because malls are private property the owner could disallow these activities. For a state that seems to be all about letting business do its business without the heavy hand of gubmint interfering, arguments that gubmint should interfere sound remarkably self-serving.