I agree with you that we must push back hard against fascist Marxist Utopianism wherever we find it. This means actively doing all we can to change the school boards and curriculum of our current government schools.
But...I disagree with you that government schools can not be shut down. Why?
Answer: Seemingly large and intractable institutions can lose their legitimacy almost overnight. For example, Martin Luther and the Catholic church, the fall of the Soviet Union, and our own American Revolution.
We now have voucher, tax credits, and charters in many states. The waiting lists for these programs are very very long. The parents whose children are on this lists are politically very well organized, vocal, and effective in their lobbying efforts. Legislators will not ( and are not) deaf to their pleas. Since government schooling is the largest expense for most counties and the state, charters, vouchers, and tax credits are one of the most effective means for balancing the state budgets and getting government school employees off the state pension and benefit programs.
Then, as conservatives, we must encourage parents to take their children out of the government schools. This starves the government schools of the funding that goes with that children.
Finally, homeschooling continues its remarkable growth of 7 to 15% a year.
Taken all together, I am very hopeful.
There are large swaths of the US population where the 'parents' don't even fix their kids' breakfast, but rely on the schools. There is no possibility that all parents in the US are going to assume responsibility for securing an education for their children outside government provided schooling.
Alternatives will probably grow in size, and leave a smaller and even more disfunctional and ineffective government system, but the government schools are here to stay.