That someone believes they deserve admission to a specific set of schools as much as someone else who is working 10-100x harder with the specific goal of attaining admission to those schools? Yes, I believe that is entitlement.
Why not have more good schools? Why not give rewarding jobs to smart talented people from any backgroung instead of people with degrees from a small number of blessed institutions? Why not give better teachers to kids who aren't lucky enough to have education-minded parents? Why not level the playing field?
Stuyvesant has plenty of bad teachers. The good schools are largely defined by getting the good students. If we switched the student body of Stuyvesant with that of Washington Irving, Washington Irving would become one of the best high schools in the city and Stuyvesant would become a school where parents beg their students not be sent.
This is the most important point. The benefit of these elite schools, is that peer pressure plays a positive role. All kids compete at these schools, form study groups etc.. Teachers want to teach motivated students and these things together drive education.
Beyond what comicjk said, yes in general we should yearn for the best education to be given to all of those who have a thirst to learn it - but that is not the point of this article so much as saying which students in particular should get into a selection of schools.