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 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.