> What matters is whether their reporting is factual or not. In this case, there seems to be little doubt that the reporting is factual; the reported lawsuit does exist and does make the reported claims.

That matters, but it isn't the only thing that matters. Again: the fact that the NY Post themselves didn't outright lie in this story does little to refute the idea that they're a disreputable paper. The way they choose facts to report and how they arrange them can be the real source of disrepute.