First, there are no prohibitions for schools in the Constitution; and, Congress in empowered to solve problems through legislation. If you wish to make this point, please quote the part of the Constitution that makes you think so, and then include supporting interpretations by the Founders or other Constitutional authority. I read the entire Constitution regularly, including the Amendments. I don't see any justification for your stance. On the other hand, I can find a great deal of sensible material related to the establishment of public schools.
Second, public schools are not established by the Federal Constitution. In every single state, schools are created by the state's constitution. The states are guaranteed that power under the terms of the 10th Amendment.
Third, public schools were instituted way back in the 1540s by Queen Elizabeth I. She understood the advantages to society brought about through general literacy, even for those who could not afford private schools. The first public school in North America was established in the early 1600s at Jamestown.
Fourth, private schools would prevent some people from obtaining even the most meager and basic education. Public schools are intended to raise the standard of living of the entire Nation at least to the point that all people can be theoretically self sufficient. A school that is organized for profit will most certainly charge what the market will bear, thus excluding many people from a basic education. How could this possibly escape your attention? Perhaps you prefer to have a serf class.
Good minds have agonized over this issue for centuries. The real problem with our public schools is that they wandered off the reservation, due mostly because of inattention and apathy on the part of conservatives. I guarantee you, private schools would also degrade in quality once parents sue them because their little darlings can't make the grade. Furthermore, degradation of private school curriculum would also occur given the inattention of conservative people.
The constitution was not designed this way. This is propaganda promoted by progressives.
There is no "schools" in the enumerated powers. The end.