Might be worth differentiating within the horse program category.
Lots of boarding schools have equestrian programs with competitive teams that ride English style (Chatham Hall, Ethel Walker, Foxcroft, Kent, George, Thacher)… Some schools even allow or encourage kids to bring their own horse.
Fewer schools have Western Riding available (Thacher, Orme, Ojai Valley School)… but only one school has a Horse Program that is both mandatory and central to the school’s approach to building character, responsibility, and self-awareness (Thacher.).