I would say that there’s correlation, not causation. Let’s change my example to Groton and Brooks, to pick two schools that are more comparable in size and feel than Exeter and Brooks. I mean no disrespect to Brooks at all when I say that Groton is filled with kids who could also have gotten into Brooks if they’d applied there but the reverse is not true. This is not to say that there’s no one at Brooks who chose it over Groton (or its ilk), undoubtedly there is and maybe that kid gets into Harvard anyway. But that’s not going to be the norm. Let’s say we have a kid from Groton and a kid from Brooks who both really, really, really want to go to Harvard. Statistically speaking, the kid from Groton has a better chance at getting in than the kid from Brooks – not because Groton gave him a superior education, but because the Groton kid was more likely to have higher SSAT scores to start with, which in turn means that he is statistically more likely to have higher SAT scores too (someone had a great analysis on here a year or two back about strongly SSAT scores correlate to SAT scores), which means he’s more likely to get into Harvard than the kid from Brooks with the lower SSAT/SAT scores. So going to Groton didn’t “cause” him to get into Harvard, but there is a correlation.
Now, whether that kid with the higher SSAT/SAT scores would still have gotten into Harvard even if he had chosen to attend Brooks instead of Groton is an entirely different question, and one that I agree cannot be answered in any way by matriculation stats (or indeed by any stats that I can think of). That’s the question currently being debated on the big fish/small pond thread, and I think it’s clear that this is one of those existential questions to which there can be no provable answer.