Yes and no. A mathematical explanation can logically reduce a fact that we consider complex to a set of simple facts we would be willing to consider as obvious. Of course, these basic facts are postulated rather than explained. In the case of the normal distribution, one could show that it is to be expected under certain assumptions, which often happen to be fairly reasonable assumptions to make about the distribution in populations of things like intelligence or height.

Probability is indeed a field that gave and gives rise to many questions regarding the interplay between empiricism and the human mind. Lumping the various questions, phenomena and effects into one unintelligible entity seems a bit old fashioned. Looking down on people who abstain from such lumping doesn't seem justified to me.