Don’t send your kid to boarding school if you don’t like gatekeeping. From the experience of parents I know, places like Choate, Exeter, etc. are pretty transparent about how college admissions work. And they are NOT going to tell every kid who would be a solid fit at Lehigh or Muhlenberg or Trinity "sure, apply to Harvard, Princeton, Yale and Dartmouth “just to see what happens”. Not happening.
You don’t like it? Send your kid to public school.