Many catholic high schools do send students to catholic schools all over the country. They all have a lot of connections to the administration and the faculty. I know parents who require their kids to go to a catholic school, so the kids look for schools where they want to live, or play a sport, or just get as far away from mom and dad as possible.
One friend (from years ago) told me “Mom and dad required a catholic school. I required an 18 year old drinking age. Marquette it is!”