That has been tried before. What happens is that people choose the school where there is easiest to get good grades. Fast forward a generation or two and there is a race to the bottom, schools take their mission to prepare for standardized testing seriously at the expense of learning. Over time, our outcome will depend on what you measure.

School systems are also special because you don't really want to overprovision school seats, so if there at the end of the day are as many seats as there are pupils in the system as a whole, there can only be selection and never competition in the economic sense.