No question the top private schools have much better counseling resources than most publics, but just looking at the number of students admitted from privates vs publics, as as a number or a percent, doesn’t necessarily tell you if and how large of an advantage going to a private is over a public.
We are looking at very different pools of potential applicants. The top privates (and magnet type publics) are themselves selective. There are just more high achieving students and ones that have proven that they can test well. If their families are already spending big bucks to send their kids to private school, we know those families already are willing and able to spend big bucks on education in general and they probably already are targeting the elite colleges. A good point that @hebegebe makes is that there is likely a high percentage of already hooked kids in private school. The fact is that top private schools have a proportionately greater number of qualified and advantaged applicants who are going to apply to T20 than your typical public school. It is primarily because of who their students are not because of the school itself that accounts for their relative success.