It is true that 'privately run schools paid by the government can provide public education'.  In Oregon that tends to be charter schools aligned with a local district, receiving tax money and subject to most of the rules applied to public schools.  They kept 'separation of church and state' but may hire non-certified teachers, which cuts their costs.  You are mistaken in thinking DeVos wants public education.  She advocates for schools receiving government money but with no government oversight for subject matter.  What better way to control the thinking of future citizens?