Your alternative is also often a much worse school. I went to CDH and many of my friends would have went to Johnson, Harding, Central, Minneapolis North, etc. There is nothing wrong with those schools, but we have to understand why a parent of a kid growing up in Minneapolis North might like the idea of their kid going to a private school.
Also, if they're a great athlete, you can point to a long list of the people who are on athletic scholarships at various schools that graduated from CDH, STA, BSM, etc.