The elite prep schools are more likely to have counselors with connections, inside knowledge, or at least sufficient historical knowledge to know which top-end students are good fits to which elite colleges from the colleges’ points of view, so that they can advise students which one would be more worth their effort applying to (rather than applying to HYPSM but perhaps not applying to some other highly selective colleges that would see them as better fits).
Students at other public or private high schools are much less likely to have this kind of information advantage when making their application lists, and will have to discover it themselves, if it is even findable by unconnected outsiders, and they know where to look.