Iacisrmma, Rav Zafrani, who was Rav Mordechai Eliahu’s assistant, said that there are only one or two in a generation who have that width of knowledge. Moreover, there has to be an assumption that the gadol is not surrounded by courtiers who control what info he sees.
Little, could it be that these schools have superior secular studies programs? Of course, this might be a chicken-and-egg questions. The students who would prefer a public school because of the secular studies programs (and BTW, even fifty years ago the specialized high schools in NYC all had some frum students – I know this for a fact as I went to Bronx Science from ’68-’71) would not be accepted into Chareidi schools. The tuition bubble is also a major factor. I personally know families who made aliya because of this.