I think you central questions it the most important one, but the dichotomy between private and public is forced. A phone company is not forced to moderate the communication of its clients.
I don't believe parent said they were forced, only that they would do it. And that makes sense, since their desire is to minimize damage to their brand.
There is a simple solution to this if you don't like the way they behave: stop using their product, and recommend others do the same.
Of course, your not using it doesn't prevent others from using it if they want to, but that's freedom of choice for you.
I am all for the free market, but when companies grow too dominant it no longer works, and we should not pretend otherwise. Just look at what happened to Parler.
Alright, I jumped the gun there I guess. Last I heard they did find hosting elsewhere because, and this is the important point, none of the companies that shut them down had a monopoly on internet communication.
I agree, its an open problem. One one hand no one elected these companies as arbiters of what is good but on the other society at large is not equipped to handle the tsunami of speech that the internet enabled. Something has to be done.