I think the NY Post is probably getting played. But the media has gotten behind plenty of stories about Trump that turned out to be false: https://www.rollingstone.com/politics/political-commentary/h.... This sloppiness is a general symptom of the need for media organizations to beat the Twitter cycle.
The question is whether this standard of reliability is one that they can actually apply evenly. This is a far higher standard that what was pitched. This content moderation was pitched as being limited to Russian bots and “vaccines cause cancer” and the like. Moderating articles from a publicly listed brick and mortar news organization, even if it’s a tabloid, based on the perceived credibility of the underlying facts, is a whole new level. It’s dramatically raising the bar. And maybe that’s where the bar should be, but I absolutely do not believe that Facebook and Twitter have the resources or integrity to apply that high standard fairly.