Yeah, I like that. It doesn't hold though, because some of us have learned to change our moral behavior based on meta-cognition. There are not that many of us, a maximum based on other studies of the ability to do meta-cognition and related self-reflection thinking and adjustment of feelings and emotion, we are at best 10% of the adult population.
I mean ,we are there, because your quote admits it itself: "...were rarely good explanations."
So doing science as one factor explanation never works when it comes to humans. Moral intuition is a part of it, but it is not all of it. I mean, I don't just intuitive moral judgments any more. Even what I do morality on the "fly" is "colored" by that I spend a lot of time reflecting on my morality and adjusting it. Yes, I do it some times, but not always.
Here is some science of that:
https://www.simplypsychology.org/kohlberg.html
Now stages 1 to 4 are generally intuitive in some sense, but stage 5 is not. You can only do that using meta-cognition and that is no intuitive.
Now I am not nice, but when it comes to human behavior never rely on only one model/theory and what not. And yes, even Kohlberg's model has limits. So do Haidt's.
Now if you want to debate morality, you can't just use Haidt. Nor can you just use Kohlberg.