Our experience with elite prep schools is that a LOT of admissions slots are given to legacy students and students whose parents are likely to be large donors to the school. Then there is likely a diversity slice and some athletes -- and a few slots left for everyone else. We applied to the top K-12 school for our kids in another city, and both were rejected (one with a 160+ IQ...) -- we heard from someone later that there were TWO slots in the kindergarten class that did not fit one of those criteria I listed above. A ninth grade class may have more slots, but it is still fewer than you think if you are "unhooked". Hate to say it, but this is good preparation for applying to the very top colleges -- there are a lot fewer slots for unhooked kids there than the colleges like to admit, too.
You also forgot to add the students of staff members of whom there are a lot.
In the "elite" school in my area, here is how it goes: if there are 100 seats for K, they try to allocate 50% of them to boys and the rest to girls to keep up gender diversity. If you had a boy, he is automatically competing for one of the 50 seats and not 100 seats. Then, those 50 seats for boys get further sub-divided between - the legacy quota, the donor quota, the staff quota, the sibling quota, the seats for the "notable achievers" (that 5 year old that played in Carnegie hall or has a sky high USCF rating etc), the quota for "financial aid" students (their charter insists on "financial diversity"), the quota for "celebrities" (founder of hedge fund, CEO, politician, potential donor, notable sportsman), the quota for kids referred to them by the school board. In reality, there is hardly a single seat left for a well performing gifted kid without any "Hooks". I was told that many long time staff had to wait for years to get their kids into their schools (staff quota was limited to a certain %). So, a PG boy (or girl) with a sky high WPPSI score or a 99 percentile ISEE score (higher grades) with no hooks might not end up with a seat at all because the real quota for such a candidate is 1 or 2 seats and not the 100 openings that they talk about during admissions events. The K entrance sounded a lot like the Ivy league admissions.
