Groups across political divides don't seem to even live in the same universe. Fact-checking cannot fix this.
Humans live in reality, but they also live in their own mythologies. Although Enlightenment ideals would suggest people care about adopting mythologies that accurately explain reality, it turns out it's much more psychologically important for their stories of the world to be perceived as true than to actually be true.
Fact-checkers codify the stories and narratives at the group level and serve as a means of enforcing that group's ideology. They are a form of ideological dominance, and they do not persuade.
To be willing to listen to a fact-checker, you must already trust the person doing the fact-checking. If you're a member of QAnon, why would you believe a fact-checker created by CNN? Institutional statements of worldview will not "solve" distrust in institutions. It is impossible for fact-checkers to bridge divides where no trust exists.
The best a fact-checker can realistically hope for is to prevent members of their in-group from diverging from their own worldview.
Fact-checkers seek to enforce narratives, not accurately describe reality. Who fact-checks the fact-checkers? Sadly, there is no ultimate arbiter of Truth.