There are various options. Since facts should rely on proofs, it should be straight forward to establish the baseline.
For the grey area, I would allow people to choose their fact provider. The providers have to document the reasoning for their 'facts'. Since time establishes which 'facts' are facts, you can determine which providers are sincere and ask users to choose among them.
It's not straightforward at all. You're being dangerously naive. The fact checkers inevitably pick and choose "facts" to support their preferred narrative.
That's an investment in disinformation that has to be made.
As you write:
>During the COVID-19 pandemic, social media companies censored many posts as "misinformation" which later turned out to be accurate
The fact checkers who pick wrong facts stand out and can be excluded. But it takes some time.
However, that only affects people who don't care about their information sources. With a central repository, everybody could check the foundation of "facts" that are used for censorship and switch their fact checker if necessary.
And who pays for the central repository? Who decides which entities are allowed to be official "fact checkers"? Why should any of that be used for censorship, and who should enforce it?
Technically, the central repository can be an almost static wiki. The costs will be very low and can be paid for as a fraction of the moderation fee.
Users choose their fact checkers. Some have to be chosen, because that's what is required by law. As such, some state agency will enforce whatever has to be enforced.
This is not a complicated idea. Create a platform and a market for the problem, and let the market find the optimum. Right now, every social network has a monopoly on moderation of fake news. That will lead to market distortions.
Who pays the moderation fee? Who runs the servers? Who decides whether someone is allowed to act as a fact checker? None of that would be enforceable. It's just a stupid, impractical idea from start to finish.
I already wrote about who has to pay the fee. Everything else can be implemented one way or the other. The important part is the idea that moderation can become a market.
For the grey area, I would allow people to choose their fact provider. The providers have to document the reasoning for their 'facts'. Since time establishes which 'facts' are facts, you can determine which providers are sincere and ask users to choose among them.