They do have systems of marking applications of prospective students approved by a coach.

This. Not outright. Not Ivy League, but I know Stanford does.

Ivies give so much need based scholarships that athletic scholarships aren't required. If someone's parents and poor, the school finds a way for them to graduate debt-free. (This is a big plus for them)

As for sports having an impact... They may not let in a 900 SAT football player, but 1200 may be enough for an Olympic rower.

That they are revenue neutral makes it even worse.