Public universities (what Americans call public schools in the context of higher education) are optional to the exact same degree as private ones. In other words they are all schools that you apply to.

They also produce more "elites" than "elite" schools do if you go by executives at F500 companies and politicians.

Are we going to pretend that Berkley, Michigan, UNC-CH, UVA etc. do not produce world class educations from world class people?