If we want any sort of free society I don't think we _can_ stop fake news. We can only attempt to counter it with reason, debunking, and competitive rhetoric. We maybe can build tools that attempt to amplify debunking, build institutions trusted as neutral arbiters or that have other forms of credibility.
For my part, I don't think anyone anywhere should be prevented from saying the dumbest things that pop into their heads; what I disagree with is giving everyone a global megaphone and then artificially removing the consequences for saying the dumbest things that pop into their heads. :)
If you have a reputation that is tied to the community in which you are participating, and your reputation affects the reach/visibility of your messages in that community, then as you behave at odds with the standards of that community, your reputation goes down, thus limiting your reach. Exactly how to implement that well is the billion dollar question, but at the heart of it all is a simple feedback loop.
That's a nice concept, but it is unclear how to implement this. If you have a low reputation, you can only reach X users, but if you reputation improves, you can reach 2X users? How do you set these thresholds? How do you pick the X unlucky users to receive the low-reputation tweets?
To combat viral fake news, the top 0.001% of users could have manual reviews, as should viral tweets that are about to explode into reaching hundreds of millions of users. Having a manual moderation queue for any tweet that has grown from 0 to 100M rewtweets in a certain time frame, as well as for everything tweeted by users with X million followers, would be quite a low cost.
We have alredy seen some of this where Twitter has manually labeled information as being false, from e.g. Donald Trump. I doubt they'd give that treatment to any of my claims to 20 followers, and nor would they have to. Because what I write simply doesn't have the same importance when I have 20 followers.
Eventually in the end, you do need the ivory tower with the ministry of truth saying exactly what is and isn't acceptable. Should Rogan's Covid conspiracy tweet be blocked? Given a warning label? Or allowed to spread? That would have to depend on the level of harm it can cause. There can't be a "marketplace of ideas" where everyone is forced to make their own judgement and where all content is acceptable. Not just because US conservatives will find it hard to tweet, or because advertisers won't advertise, but because it'll have terrible signal to noise ratio for anyone, causing it to be pretty deserted.