Interesting article that identified one of the problems in public school systems that is often ignored and that is the weight of non-teaching staff. This is normally hidden in offices with titles such as curriculum support. If these support staff were in the schools they could provide useful pedagogical services but they are usually footsoldiers in bureaucratic turf wars.
Unfortunately I was waiting for the answer but didn’t see any. I even went back to reread the last paragraph. The problem with vouchers has been well argued. So his will that improve the public educational system? High quality private schools already draw the best students from successful families and, as you note, charters are supposed to be the option for choice but tend to be no better than standard public schools or are at capacity for education as we currently do it. So how does what you suggest not make it worse?