In regard to primary and secondary schools, my opinion is that education should not be entirely privatized. There are already many private schools at these levels and families may choose to send their children to them if so desired. They claim, and may well possess, higher standards of education (or add religious addendums to the standard curriculum), and typically reflect well on applications to elite institutions. The problem with applying this model to the entire system is that running a school is not a cheap affair. This is reflected in the cost of tuition at private schools, with some ranging into many thousands of dollars per semester, depending on the quality and prestige of the school. Most families below upper-middle class financial status simply could not afford even the less expensive private schools, and likely couldn't afford to take on loans for it either. If this were the case, we'd return to a state where the more wealthy were educated and could perform jobs which required it, while the rest were left to learn trades and do menial labor while forgoing the basic pillars of reading, writing, arithmetic, and so on. This worked for a time when many tradesmen were needed, but the problem is that there are very few trade jobs left relative to the population (example: approximately 1% of the population now produces enough food for the rest and surplus via farming in the US). You can see that this would basically cause a collapse in the current economy and society as unemployment quickly reached ridiculous levels.
As for post-secondary education, the story is the same.There needs to be some sort of (at least partially) publicly funded options as the trade and labor jobs continue to decline and higher degrees continue to increase in importance, or there will be severe consequences. This may be more than you were looking for, but I hope it helps.