On the conservative movement level, I don't think that conservatives should be surrendering the school system to the liberal establishment. The progressives were more strategic than us a couple of generations ago and began to infiltrate the public school system. Conservatives need to retake the public schools, not cede them forever to the progressives. It's going to take conservatives a long time to do so because once ensconced, the progressives are not going to passively abandon their position.
On an individual level, how you school your children is a personal decision that each family needs to make considering their own situation. My three children graduated from a private, Christian school. However, that doesn't mean that we should surrender the public school system. Several of the graduates of that Christian school have strategically taken jobs in public education (or are planning to) to fight the social war on the front-lines. Such actions are exactly what are needed to re-take the public schools but it is going to take a generation or two to see major successes.
Conservatives taking jobs in public education is not enough.
The Dept of Education and the teachers unions direct all the curricula.
If any teacher strays noticeably from the curriculum, that teacher will be fired.
The teachers unions are run by global elitists.
You won’t be able to displace the union elites with conservatives.
The Federal Department of Education provides the carrot (MONEY) to the union’s stick. Good luck draining THAT swamp.