I agree to an extent. I for one was very distressed for a while when I started to believe that moral realism is false.
However, this is not exactly a secret matter. Moral realism and anti-realism have been discussed openly for a while. It may have caused people to act less morally, but I am not sure. That is difficult to measure.
For me, I do not think it made a huge difference on how I behaved. I struggled with cognitive dissonance, and still do, but my conscientious urges remained in place.
I gather that you disagree with the Psmiths’ conclusion.
Can you think of any examples of similar shared fictions that are much more unstable?