On that I'd have to disagree with you. Students who went to Stuyvesant High School or Bergen County Academies or any other really, really prominent public school get just as much recognition as students who went to Phillips Andover. Honestly, I think you're just kind of isolated from the appreciation of that sort of recognition.
I went to the 11th ranked public high school in the state and we get no special recognition. The Pingry school, which is in the same town, is held in much higher regard than my school. Same situation with Delbarton and Immaculata, which are both very nearby.
Pingry School? Who? I think you're overstating the local prominence of private schools as opposed to prominence where it matters, on the national stage (and in the minds of college admissions officers).
I live in a cluster of some of the wealthiest, most power-obsessed, label-loving towns in the country. All anyone talks about all through high school is how we compare to each other, other high schools, private schools, and how that applies to what colleges we'll get into. You can go down any street in all of my area and count maybe one car out of ten that doesn't have some kind of nationally recognized University plastered across it. We are obsessed with image and prestige, moreso than I've ever seen anywhere else I've been.
I'll keep that in mind next time you're going on about how great New Jersey is.