Your website isn't a public square because you're the one publishing content to it. We're discussing the public platforms that Facebook, Twitter and Google run, and I don't consider those three companies as publishing the content they host.
Social media does, in my view, publish the content they host. It's not like the telephone, where you establish a direct connection to your audience and the mediator gets out of the way. When you post something to social media, there are three separate steps that happen:
1. You send the content to the SM company.
2. The SM company does some kind of processing on the content.
3. The SM company publishes that content (or not) somewhere on their site.
These things happen pretty much instantaneously, but they are still happening. Posting to Social Media is more like writing a Letter to the Editor of the newspaper. They receive the letter, decide whether to include it in the paper, then include it in the paper.
What you permit, you promote.
Don't you consider it publishing when they express editorial control over what topics are allowed on their platform, without transparency or accountability of what the forbidden topics are at any given point of time? They also modify the user generated content and add labels and "fact checks" which are prominently displayed.