This wouldn't work, at least not outside of places like HN with a strong technical bent and a strong assumed knowledge base of the people.
I've done fact-checking work professionally; it's not that fact-checking itself is difficult, it's that fact-checking can only work as intended in certain information environments, and mass media is NOT one. I've also studied filter bubbles, algorithmic influences on political POVs, etc.
Your proposed system might work in a vacuum, but unfortunately, our modern landscape is NOT such a vacuum.
Issues with your proposed system:
2.) What counts as a claim? Is 'the sky is blue' a claim, or is that common knowledge? We clearly wouldn't flag common knowledge because using the flag when it's not necessary deprecates the usefulness of the flag, but then who decides what counts as common knowledge, particularly across different cultures and countries?
3.) A common metadata way of judging someone online is through their sources. Think of many subreddits which don't allow certain 'left-wing' or 'right-wing' sources. The supporting text will be upvoted and downvoted based on the readers' idea of the source: In the politics subreddit, a NYT 'claim support' would be upvoted while a WSJ one would be downvoted. Instead of using headlines as proxys, the URL would be: "Oh, it's X. They just always lie. Don't even need to check; downvote, nobody listening to THEM could be correct."
4.) Once the whitelisting and blacklisting go into effect and people are aware of it, the voting becomes even more distorted as certain people with grudges can both climb community hierarchy (and therefore put themselves in positions of influence and say things like 'we don't read X here, they're all liars') and encourage voting en masse to blacklist certain sites they disagree with.
5.) This is terrible for search and archiving as well as for tracking bad actors.
And that's just off the top of my head.