Privates are self selecting so have much bigger range of hooked students, including really smart URMs, and high performing athletes (not just legacy and Uber rich). Plus academic super stars. Thats a big reason why they disproportionately have more kids at top schools. From my experience the standard wealthy private school student is not getting into ivies no matter what their parents influence. However the very good URM especially with athletic prowess is getting in.
This was not the case, years ago.
Many kids at top privates are hooked. Not the case at most publics.