It isn't nihilistic. It is honest. We do owe it to ourselves to acknowledge that reality continues to exist. So in keeping with that, it makes sense to acknowledge that in reality these fact checking websites can be compromised. They aren't a bastion of truth, even though they claim to be.
In my personal experience, when fact checking political fact checking, three out of the four claims I investigated turned out to be highly misleading. Usually it was by the person doing the fact checking leveraging the fact that people wouldn't have full context.
I'll give an example. Unemployment. Someone will say un-employment is higher then the statistic suggests, because the way the statistic is tracked has people who are unemployed for a long period removed from the listing. They will then give an estimate of the real statistic. The fact checking will say that this is a pants on fire lie, then cite the unemployment statistic. They will not address the central claim about how that statistic is recorded.
In my estimation, these sites are largely propaganda. Or maybe I'm being too bold in saying that. I know there is research that suggests that people are literally less able to read statistics and think factually in the presence of their political opposition. If given charts to read, charts that disagree with political positions are more likely to be misread then those that don't. There might be more at play then an attempt at influence leverage on the part of the fact checkers.
Either way, I'm inclined not to set the prior the current iteration of fact checking websites are an example of striving for truth and so deserve special value when updating what I believe.
However, I tend to find that non-political and not hotly debated topics don't seem to have this problem of being frequently wrong. So when things aren't highly political, I probably will allow the use of fact checking, because if I'm striving for truth that is expedient.