I have studied at an Ivy and I can tell you that most of my fellow students, and many more with whom I've worked since I left, are not very special academically. But more than my personal, anecdotal, evidence, this chart, from an NBER working paper, demonstrates my point.
At MIT (which is not an Ivy), the probability of admissions increased monotonically and steeply as SAT scores went up. At Harvard, the probability of admissions increased until the 92nd percentile, then plateaued until the 98th percentile before increasing again, although much less steeply than MIT. Princeton's was the most interesting. Like Harvard, the probability of admissions increased until the 92nd percentile. However, it decreased until the 98th percentile before increasing steeply. The interpretation in the paper was that Princeton, in those days, expecting to lose most common admitted students to Harvard, was "ducking" students whom Harvard was likely to admit, in order to protect their "yield". This interpretation was subsequently validated in an interview with a former Princeton admissions director.
At MIT, a student with 98th percentile SAT scores was 4 times as likely to gain admission as a student with 92nd percentile SAT scores. At HYP, a student with 98th percentile SAT scores was no more likely to gain admission than a student with 92nd percentile scores.
All these schools are more likely to admit the very academically exceptional students; they're all happy to produce some professors. At MIT, a student with the very highest SAT scores was 5 times as likely to gain admission as a student with 92nd percentile SAT scores. At HYP, a student with the very highest SAT scores was only twice as likely to gain admission.
SAT scores are not a perfect proxy for academic ability. But, if anything, their known biases towards well-resourced students who receive training will cause them to overestimate actual academic ability in this part of the population (top decile).
Btw, three of the co-authors of the paper were affiliated with Harvard and Penn, so they had no incentive to rubbish the Ivies.