Southlake Carroll's situation isn't like private school if they simply pull kids from their own district. That is pretty standard operation for public schools, unless they are short students in the district, then they can be classified as "school of choice", where people outside the district can enter a drawing to be selected to attend that school.
Michigan has big money public schools as well in affluent areas, but the private schools are ridiculous we have here. Not sure if you've ever seen the movie "8 Mile" with Eminem. In the last rap battle he does, he says the line -- "But I know something about you, you went to Cranbrook, that's a private school". It cost more to go to Cranbrook High School (just under $60,000 a year) than it does to go to the University of Michigan ($53,000 a year).
We had to play Cranbrook in multiple sports when I was in high school -- they'd have foreign diplomat kids, they'd have kids who came from royalty, then you had the who's who in Michigan and all their kids went there. Their HS campus was unbelievable -- everything state of the art.