Just to correct your son’s impression, @MilouForever:
https://www.nyu.edu/admissions/undergraduate-admissions/nyu-facts.html
59% of NYU’s freshman class went to U.S. public high schools. 27% went to private schools or parochial schools and 13% are internationals! I don’t recall anyone from NYU ever saying that the GC recc was important in our admissions meetings. My (NYU sophomore) D’s GC recommendation was probably fine… but her teacher reccs were likely much more personal and detailed.