If private, the owner can do whatever

So can Facebook, Twitter and Google. Yet Conservatives are actively campaigning against it.

No one who hates Marxism would pay to learn it, now, would they?

Seems like a lot of people complain about it, even if they are in private universities.

False accusations can be labeled fraud. Telling the police or not doesn't matter when damage can be measured. Free speech doesn't mean lack responsibility.

We already have something to deal with it. Defamation laws. But if you open the door for the government to police lies on social media and jail people for accusing someone of a crime, you will get people in jail for calling someone a whore in a province/state where prostitution is illegal.

are meant to actively break down Western society

Still free speech.

lead to doxxing

Doxxing is free speech.