It's a common defense of the SAT that it gives a fair shot to smart students whose cultural background means they don't know how to play the admissions game.

But I think this is not true at hyper-selective institutions, because as you say a perfect SAT score is necessary but not sufficient.

A student who has been on the normal, non-advanced educational track, not playing the resume game, and suddenly gets a 1600 SAT, still typically won't have a shot at MIT.

What they will get is a bunch of great full scholarships to good, but not hyper-selective elite, universities.

Absolutely. MIT wants to admit a varied population of students who are probably smart enough to do well if they apply themselves. They don't really need a harder standardized test. (There are also whatever the Achievement Tests are called these days.)