She's a billionaire and her and her family have never attended or paid for public schooling (wrong of course since she has paid her taxes).  She has no experience managing education.  Both comments relate to a person you do not know except via media indoctrination.  She has not proposed any specific policy in her new job, you fear only your speculation.

Public education means tax subsidized education, not just the state-owned system we have now.  Today, unions rule state-owned public schools.  The best interests of the unions are served.  A public education system that pays for student's education, but allows privately owned schools to compete with state owned is not clearly inferior and could possibly be superior to the system currently in practice.  Competition generally sharpens performance and product quality.  Yes, today's union teachers would hate competition and will fight to the death to prevent anyone from rocking their boat.  But her ideas may benefit the students.