You misunderstand. The government pays for the education and private concerns administer it. No unions, no bloated administrators as we have now, just a tough curriculum administered by teachers held to strict standards and whose pay is merit based. WE could save BILLIONS while giving thew children a much better education.
And how do you propose to hold private concerns accountable?
Say for instance that Company A is administering all of the schools in the Hypotheticalville's schools districts. (As would likely happen, we live in an age of corporate consolidation, not in an age of small businesses competing with each other in a model of pure capitalism)
Now, say that Company A decides to operate these schools at the lowest possible cost in order to maximize their shareholder return, thereby providing an exceptionally low quality of education for their students.
How does the community hold Company A accountable? There's no competition, no way to take your children to another school without paying an exclusive private school a great deal of money in tuition.
The parents therefore are as much of a captive market for Company A as they were for the public school district that existed before them.
The point I'm making is that businesses do not exist for their customers, they exist for the benefit of their shareholders and therefore they do what makes economic sense in order to maximize shareholder return.
Therefore, placing businesses in charge of education ensures that education will be delivered in a manner which maximizes the benefit, not to the children being served, but to the owners of the corporations delivering the education.