FWIW most of these day schools that you are dismissing as segregated are much more multi ethnic and multi cultural than most suburban high schools which are segregated based on the way we live in America today which is largely segregated. The elite NYC schools have made pretty incredible strides in diversification and although it is absolutely (and obviously) true that they can not offer the geographic diversity of a BS it is false to suggest that they are segregated. They have large endowments and FA budgets and they use them well and with good intentions just as BS do