The parent's point, I believe, was that it is privately owned, rather than being an instrumentality of the public/government/state/people, regardless of how it's called.

There's an argument to be made that even privately owned entities are to some degree still stewards of the public trust and therefore owe some duty to the public, especially if they benefit directly or indirectly from public subsidy, and/or if they are able to exert monopoly power. This is generally expressed through regulation.

There's another (albeit related) argument to be made that if a platform for speech[1] is provided to the public, that it should conform to the same standards of freedom as elsewhere, including government. I believe this is what is mainly being debated here. Attempting to comply with the spirit of such an argument can be challenging for a company whose platform spans multiple "elsewheres", but there is also an argument in favor of greater freedom of speech, no matter the local standard (which I personally happen to support).

[1] or, perhaps, anything, such as a space for gathering