Many of the private schools on Long Island benefit from well to do families that live in towns that have diverse ( both financially and ethnically) communities. Like Huntington, Bay Shore,Amityville, Bellport etc. These towns have areas near the water that are absolutely beautiful and areas that are absolute ghettos.
The education may be a little bit better but not enough to warrant a kid from CSH,Garden City,Syosset, Smithtown etc to deal with the commute and extra expense and hassle.