Some want prayer in school, some want condoms. Printing prayers on condoms satisfies nobody.
In all countries, in all centuries, the primary reason for government to set up schools is to undermine the politically weak by convincing their children that the leaders are good and their policies are wise. The core is religious intolerance. The sides simply change between the Atheists, Catholics, Protestants, Unitarians, etc., depending whether you are talking about the Soviet Union, the Austro-Hungarian Empire, America, etc. A common second reason is to prepare the boys to go to war and the girls to cheer them on.
The Left/Right scale is a misleading way of comparing political systems. It doesn't measure anything.
Charter schools are just public schools on a slightly longer leash. A dog on a long leash is still a dog on a leash.