The Establishment in the beginning of New York's Industrialized Period was White Anglo Saxon Protestant. We all live a reasonably comfortable existence now but think about the birth of the modern period. Most of America was agrarian. Shitting in outhouses, deathly afraid of Polio, unwanted pregnancy, living by gas lamp (if that)etc. The British elites, were the fountainhead of comfortable practicle worldly living. And the lower class whites and EVEN lower "browns" dreamed and schemed of getting there.
Of course what worldly knowledge existed at the time had to be transferred to the young white gentlemen of privilege so that they could assume the rights and duties of this brave new empire. Even though the actual "facts" of the world were the same, education is so much more than mere numbers and measures. It's manners, temperment and socialization. The well to do couldn't send their kind to 8 hours daily with the ill-mannered and rude "common man".
The more "waspy" private schools were Choate, Andover, Collegiate, Brearley, Dwight, Dwight York, Fieldston, Rye Country Day, etc. Dalton became the "jewish alternative".