And what exactly is the difference in your mind between "moderating" and "censoring" here? They seem exactly the same to me: both involve some other party deciding what is and is not acceptable to be published.
Private companies that allow you to reach millions of people are offering you the ability to use their service. It's not censorship when they don't allow you to use their service (and until today, they haven't stopped much), it's terms of service.
If I invite you to my house and you eat my dog's food, I have the right to tell you to leave. The movement wouldn't be what it is today without YT, so this censorship stuff is too much.
Censorship is NOT just government action.
I've long seen it used to refer to, for instance, TV network censors. And every online dictionary I can find allows for entities other than government to be referred to as censoring.
They run a platform that practices censorship. It is legal. Trying to say it is okay because they are a private entity doesn't wash and they should be called out. We could even band together and censor them.
There are several problems with this, outlined in Clarence Thomas's recent concurring opinion in Biden v. Knight. Dominant communications platforms are essentially part of the public square, and two well-established legal principles could come into play to restrict their actions: public accommodations and common carriers.
There's the additional problem that government is influencing how social media companies police content, not only indirectly through fear of retribution, but even directly. One instance that recently got media attention:
https://www.realclearpolitics.com/video/2021/07/15/psaki_wer...