"Converting the sinner" and "instructing the ignorant," however, are Spiritual Works of Mercy -- in a Christian context. "Instructing the ignorant" doesn't mean teaching them to read and to master subjects like mathematics and geography, it means educating them with the explicit intent of passing along the timeless truths of Christian doctrine.
So there is absolutely no need to spend a moment of my time or a dollar of my money on any non-Catholic who wants to take advantage of the opportunities offered by Catholic schools in education regarding secular subjects -- while openly rejecting any religious aspect of a Catholic education.
Though American Catholic schools tend to focus only on vocational training and the bare minimum of ethical behavior, that's a reason for improvement, not abolition.