I didn't miss your point, that's why I put "public or otherwise" in parenthesis.
The academic demands of public kindergarten are typically much lower than those of private schools, so a child who attended a good private pre-school should be better prepared for either a public or private kindergarten.
Is there some specific difference that you are alluding to that requires a different type of preparation for public K?
More a question of indoctrination rather than academic demands.
Coming from New York City as I do, I can tell you that PC was alive and well 40 years ago (though not called that).