Twitter the company isn't supposed to be a town square, you are correct. Twitter the product is, though. Whether that's how you see it or not, is a separate question, but those two positions on Twitter aren't mutually exclusive.
This makes as much sense as saying a United Airlines jet is a public forum, and your rights are being trampled on if you can't take a shit in the aisle.
If you assume that either everything is supposed to be a public forum or nothing is, then yeah. Otherwise, no.
Both are private property, yes. And you are correct that twitter legally has as much right to do whatever they want with their product and restrict it however they want as United Airlines does. That's not the point being made here at all and has nothing to do with the discussion at hand.
The parent comment I originally replied to was saying "how can twitter even he a public square if it is a private company, that's a contradiction". And I was just saying that there is nothing preventing a private company from having a product that behaves like a public square, hence that isn't a contradiction or an oxymoron.
The government is meant to serve the public good. The workings of the government are oft opaque, and may need to be concealed, in the pursuit of that public good.
Yes that's the entire point of contention here. Twitter is a company, its management can choose to treat it more like a town square or more like a closed network with stricter content mediation rules. Elon wants it to be more like a town square.
Arguing that Twitter ought to moderate itself according to free speech principles isn't denying Twitter's right to free speech. I'm not going to stan Musk specifically, but this is a common misunderstanding (the general formulation being something like: "criticizing someone's free speech violates their free speech!", which is patently untrue).
