Let me try my hand at this one.
Let’s say public education vanished from the face of the Earth today. With the current schools in existence, only the rich elite would have access to education, resulting in a huge drop in overall academic level.
This scarcity of educated individuals would cause businesses to pay much more for an employee with an educational background. Banks would see granting loans as a safe investment of their money, because they could be confident that anyone with an education would be able to find a job.
Businesses know today what colleges correlate to what levels of education; universities know the high schools. In a privitized system, the best institutions would be able to charge the highest fees. The schools with inept staffing and poor management would eventually disappear and be replaced with better and better institutions.
Because of the value of an education, the best individuals would have even more incentive to become teachers, because the best schools would be competing for their presence. Schools with large classes would be able to charge less, but if small classes really are better, people would be willing to pay for them. In my experience, it’s most often the public schools being forced to settle for large classes and lower quality proffessors anyway (this is not overly biased, I attend a public college).
The problem with abolishing public schools is the status quo. If there were no public education in this country, businesses could go overseas for employees. Their governments, through force, would have been able to provide them with the education that we were too virtuous to steal for them. The only environment that can support many Objectivist theories is one where use of force is equally prohibited for everyone. Anyone who uses force will have a distinct advantage over anyone who doesn’t.