My guess is the classical Christian schools are established for all the same right and wrong reasons as non-classical Christian schools. But Doug Wilson is quite clear in his speaking and writing regarding his rationale for Christian schooling, and here too, Reformed thinking is evident. These schools place great emphasis on not usurping parental authority, especially that of the father, who is viewed as the leader in the home. Parents are welcome and quite involved with the school. The curriculum is seen as the key to Christian education, and God's absolute truth must permeate all.
Sound doctrine creates sound thinking.
This emphasis on Christian thinking again reflects a tilt toward the doctrinal position rather than the Kuyperian position within the Reformed camp. There are exceptions, however. He is primarily proactive, rather than reactive. But he also has virtually nothing good to say about government schools, which I think is unfortunate, since the antithesis to which he is committed cuts through institutions, not around them. Even government schools do some things right, just as Christian schools do some things wrong.