Perhaps it can be said that Caltech has the privilege of being able to sort out the most mathematically inclined from the less so because it receives so few applications.
On a side note, some secrecy in the "admissions formula", so to speak, may be good. Since as soon as an "admissions formula" is released, quite a few parents are going to pressure their kids to fit in such "admissions formulae", without letting such kids develop in other areas. The MIT blogs say that they look for fit, however it is defined, but at least keeping the "admissions formula" secret will prevent parents from overstressing their kids to get into top universities.
Caltech's admissions are incidental to this. Many people don't apply to Caltech. But yet, a parent who looks closely at Caltech's admissions will be able to discern the factors that would greatly increase his child's chances to Caltech - the AMCs/AIME, the research, the passion for science per se (whereas many parents realize that there is no such formula at MIT/Ivies - and consequently don't have the option to load up their kids with so many extra-curriculars and empty activities/achievements). It's probably a good thing that not many people are interested in Caltech - since if Caltech's admissions policy was broadly applied to all of the top universities, well, we'd see an upsurge in the number of students who stress out all over the AMCs/AIMEs, research, olympiads, etc. (instead of stressing out all over getting their kids into as many extra-curriculars as possible).
A lot of the top universities do have influence over the minds of many parents/children - in that if they change the admissions policies, they may also be able to influence parents to make students conform to such admissions policies. But if these universities, instead, influence parents in a different direction - that is, that there is no set formula in admissions policies, it may at least make these parents somewhat more relaxed towards their children (though perhaps not for the SATs/GPAs).
There is a lot of stress over SATs/GPA, but this stress is fueled by the admissions criteria of nearly all universities - state universities included. This stress is somewhat separated from the pursuit of prestige - since there is the possibility that one without good SATs/GPA could end up a "failure in life." At least the top universities could reduce the amount of stress all fueled into empty extra-curriculars that students pursue only for getting into top universities.
Since Caltech's admissions is unique (and it has such a privilege since all the other top universities have different admissions criteria - and since math/science isn't the only thing that is valued in America - unlike a number of countries that want to become first-world countries - India, for example. Oh, the stress that would come from a similar institution in India/China), at least it can spare itself from the pressure that is mounted all over the other top universities. Now, there are other countries where admissions is purely based on "stats" on exams - China, for example (there are other countries where students are actually ranked according to exams, which are the sole way that students can get into universities). I'm not aware of how crazed out the students in China or those countries are though - but the attitude that many Chinese immigrants have over their children may be a symptom of the craze in China over examinations. It may be that this whole "college admissions" craze is somewhat exaggerated - since it seems that most of the top students and parents are fairly relaxed about the admissions process. "Most" still allows for the possibility for a large number of parents/students to obsess themselves over the admissions process.
While one could say that the pursuit of prestige is inherent in human nature (and in perhaps the nature of many social animals as well) - hey - the top humans in primitive societies got to mate more =P, most humans (and social animals) are acutely aware that they can't be at the top and accept their roles as such. However, a number of those who believe that they have a reasonable shot at prestige will start preparing themselves as such (environmental cues). A certain number of people have the personality to adopt such environmental cues (the so-called type A personalities), while others aren't as so fixed upon adopting them, even with exposure to such environmental cues. A number of psychologists do say that depression is an extreme form of those who accept that they can't be at the top - though this model is somewhat simple
http://talk.collegeconfidential.com/.../t-125007.html