All private and public top schools set admission targets and hit them every year using modeling and adjusting various levers.
There are a tiny number of seats at elite schools and that's how they remain the elite schools.
Source: Local state university in an interview. This came up during the issue of affirmative action. They pointed out (with actual statistics) on how most of them have open admission. The context was that admitting someone via affirmative action was not depriving anyone of a seat.
This was for "regular" undergrad admission. Grad school/business/law/medicine (perhaps pre-med) may be different.
All of these schools have quotas and admission targets and effectively raise/lower the bar to hit their targets.
Just look at the numbers for yourself.
Most of them admit an almost identical number of students each year. The admission rate is what changes.
This is only true for a few elite universities.
In particular, for public universities, the vast majority, including many/most of the top ones, do not have any cap on the amount of incoming students. Whoever meets the bar gets in.