Any attempt at fact checking will not be neutral. You can fact check obvious hyperbole, you can fact check because of slight misinterpretations, you can decide NOT to fact check. As soon as you decide to do it, you open the door for bias or accusations of bias.
Yes facts are objective, but when you decide to fact check or not fact check is completely subjective. Fact checking statements out of context can be misleading in themselves, and fact checking statements that were misinterpreted by the fact checkers is also influenced by bias.