Saying "both sides can't answer this one question" as though that implies that both sides are equally invested in magical thinking is a bit unfair.
Anyway, if we start inventorying our personal beliefs and the rational, empirical basis for each of them, I think we all probably practice magical thinking to some extent.
I imagine some perfectly rational, omniscient alien interrogator questioning my most deeply held beliefs about, say, free will, or my individuality, or why the world is the way it is, and placing me somewhere on a continuum that includes witch-burners and cavemen, rather than in a completely separate category.
OP said secular humanism and religion were indistinguishable - my point is they are distinguished by their fundamental attitude to evidence.