Sometimes they'll do it for a kid who needs another year of school for health reasons (terrible grades b/c of a prolonged hospital stay, etc), or just for a kid from a stratospherically wealthy/important family who had a behavior issue or academic issue their last year. (This is not wealthy/important by DC standards - think internationally. Senators are a dime a dozen.) And these are still probably the lower tier boarding schools that will do it.

Exeter does PGs mainly for athletics but not exclusively. There's usually a small number (fewer than 5) of non-athletes--mainly international students, eg the odd Saudi prince.

In terms of how easy it is to get in, it depends on how good you are and what your story is. Hockey PGs in particular are prized. I know that for Exeter's peer schools the story is much the same. In my head, Avon's entire varsity was PGs but not sure if that's still the case.