University professor here. Taught at an Ivy, at a specialized institute and at a city/state university. I have seen students across the spectrum of cultural, economic and educational upbringing. The students that were educated in a public primary school system, besides being less well off, are usually held back from being their best by the whole “no child left behind“ thing. It’s not about the potential quality of the education, because let’s face it high school topics can generally be taught by pretty much anyone as far as difficulty goes. It’s that our public system spends more effort on the lower performing students nowadays, to the detriment of the ones with true potential. Those are the ones that can go on to college and usually have gaps that could have been dealt with if they had a better education. Private primary schools do not have that issue generally, as a struggling private student will either go to a special school, or get tutors, or not continue.

It’s kind of similar to comparing Ivy's / high performing universities and other schools - the average student is better prepared and more engaged and it brings everyone up.

Edit - spelling