"these platforms are for the purpose of public communication"
No, they are not. They are built for the specific purpose of gathering the personal details of the people using them, in order to sell that information in aggregate to any advertising company who can pony up. The fact that you might be able to communicate with somebody else using the system is purely to entice you to provide that info. It worked, too ... but note that NOWHERE does that service promise to provide that capability to you for eternity. Quite the opposite, in fact ... most (all?) say that they are free to terminate their service to anyone, at any time, for any reason (including no reason). It's in the fine print of the Terms Of Service you agreed to when you signed up. You DID read that fine print, and understood it before agreeing to it, right?
"generally because at the moment it's mainly the right getting censored."
Nobody is getting censored. Only a government can be a censor. The opinions of those who get kicked out of any online service can easily be found wherever they wind up, they are not shut up, they are still freely allowed to voice their views. In extreme cases, they can even buy their own platform if they like, and talk away about anything, whenever they like ... see the former orange idiot-in-chief, the idiot formerly known as Kanye, and the stoner.
"I am confident that if one of these platforms started censoring the left, many of those who have been saying 'private business, they can do what they want' would suddenly change their view on that!"
I am absolutely certain that you are quite correct. I am not one of those people.
To me this isn't a freedom of speech issue. It's a freedom of the press issue. And as we all know, freedom of the press belongs to those who own one.
I own a printing press (two, actually). I am allowed to take contracts to print anything I like, for anybody I like (obvious exceptions exist: kitty pR0n, national secrets, etc.). But it's my press, so I am allowed to reject any contract if I don't agree with the message I am asked to print.
However, I am ALSO allowed to accept a contract I do not agree with, if I need/want the money. And I am free to reject a reprint of that contract, or further work from that party, if I no longer need the money. Or simply because I don't want to. Completely capriciously.
It's my press. My power over its output is absolute. If I refuse to print xtian or nazi or MyLittlePony propaganda, it is not censorship. The person requesting it can go elsewhere to get it printed. Or they can purchase their own press and print it for themselves.
Anybody who doesn't like this is free to not contract with me to do their printing. They can even ask their friends to not use my services. But they can NOT force me to print for them. The law doesn't work that way.
Only a government can attempt censorship. Even they usually fail.
 For values of "service" roughly equivalent to "I turned out the ram to service my sheep".
 In 1941, Norman Woelfel published a work called “The Fourth and Fifth Estate” which included the following quote: "It is foolish to assume, because in America we do not have an official propaganda agency dictating what shall be broadcast, that American radio is free. Like the press which is free for those who own and control it, the radio is free for those who can buy equipment, hire technicians and talent, and secure profitable advertising contracts.". Today, I rather suspect he would have included "Internet Servers" in that sentiment.