The thing is, "its not right" and "its not within Facebook's right to free speech and is instead a violation of someone else's free speech" are two different things.
The people arguing "its a private company" are generally not responding to arguments that Facebooks moderation decisions are merely wrong, but to arguments that they are violation of others' free speech rights rather than exercises of Facebook's free speech rights.
Which is why I linked to the moat and bailey argument. The first amendment is a non-issue. Freedom of speech is something completely different which has nothing to do with one law from 3 centuries ago that limited the power of the federal government to limit peoples rights to free speech in certain areas.
We know the opinions of the people who wrote the first amendment and they would have not been amused by the arguments we see today. These would have let the East India Company silence anyone who complained about their tea in the areas they owned a charter.
There are different understandings of the right to free speech, but a very common one (both now and at the time of the Founding and the First Amendment) is that it is the right of private parties to decide what ideas to expression, including deciding what ideas to use their resources to express and relay, free from government intervention, and that there is nothing outside of that freedom from government intervention included in the right. A private entity can, in that view, violate the right only when it acts as an, de facto or de jure, agent of the state.
It is not a bait-and-switch to hold a different view of “free speech” than your idiosyncratic one.
To put it in the melodramatic language of the 18th century: A corporation has no soul to save and no body to punish. It has no natural interests and no natural rights. It is a legal fiction whose only use is to protect its owners from ruin and whose only justification is the public good. To pretend otherwise is an affront to both common sense and god.
No, the above is the fallacy of equivocation. “Private” in “private party” unambiguously refers to the “non-government” sense of “private”, where the “party” may be a corporate entity or natural person. It does not refer to the “private(ly held)” vs. “public(ly traded)” distinction regarding how corporate stocks are traded.