The reason that centrists support private censorship now is very simple:
We are now facing disinformation campaigns that have apocalyptic consequences, and normal legal channels are quite ineffective at battling them.
"Well of course, it's not the government's job to decide between true and false, nor to enforce it!" you say.
Bullshit, I say.
Slander is illegal. False advertising is illegal. Fraud is illegal. Defamation is illegal. Perjury is illegal. Filing a false report is illegal.
There are a plethora of torts and laws where you will get sued or jailed for lying about something important (like a court matter) and in those cases the truth is a defense! That means a court decides what is true and what is false.
So we've admitted that we're willing to abridge free speech to protect truth. That's now established. But we're only willing to do it for cases where somebody can show clear damages, and where there's a clear target to sue.
A diffuse, widespread misinformation movement against a concept and not a person? Like climate change, or vaccines, or covid, or Judaism at large? Those aren't protected at all. And why not? Because there isn't a shareholder of climate change who can show that your misinformation has unjustly damaged his share price?
So given this legal vacuum, is it any wonder that the reality-based community has embraced private censorship? Nobody loves this solution, but the alternative is letting one third of the population doom the other two thirds by obstructing the kind of actions we need to survive.