And it is stll not an objective moral fact in the sense of Moral Objectivism, in the sense of a first-order fact that makes some moral propositions mind independently true.

I don't think that's a coincidence. Whether there is some kind of factual (e.g. biological) base for morality is an interesting question, but it's generally a question for psychology and science, not philosophy. People who try to argue for such a factual basis in a naïve way usually end up talking about something very different than what we actually mean by "morality" in the real world. For an unusually clear example, see Ayn Rand's moral theory, incidentally also called "Objectivism".