I'm not so sure. Many (most?) of "fact check" websites rapidly lose my trust, because they can't help but editorialize unambiguously true or false facts because they're being employed in a way the authors of the website doesn't like. Calling oneself a "fact checker" puts the expectations of accuracy and lack of bias much higher, and not many are capable of refraining from editorializing.

One concrete example that came to mind is some political that cited a study that found open-carry states had some percentage (IIRC ~17%) fewer homicides. The political cited a specific figure from the study, and it was the correct figure. Politifact rated it "half true", because it found this to be a naive interpretations of the data and ignoring the fact that open carry states tend to be less densely populated and urban centers tend to contain the most violent crime.

I agree that citing these statistics and concluding "Study X says open carry states have less murders, therefore making a state open carry makes it safer" is a naive interpretations of the facts, but the facts are unambiguously true. Labeling this fact as anything other than true goes outside the realm of facts and into the realm of interpretation of the facts. The publication would have have been better off correctly labeling the statement as true, but explaining that simplistic interpretation of facts is not effective.

Even the "fact check websites" have their biases, usually choosing to lie by omission and by presenting conclusions. They aren't just "here are all the facts in no particular order so you can make up your mind". They are "here are the facts I've chosen to highlight and this is the conclusion to arrive at from these selected facts".