Regarding this statement, this is generally true, but not exclusive to “top private” colleges. Diversity tends to increase as one goes to higher grades in school. I.e. middle school tends to be more diverse than elementary school, high school tends to be more diverse than middle school,. and college tends to be more diverse than high school. This is simply because the higher grade level schools draw from larger regions that have more diversity than smaller regions like the attendance zone for an elementary school.
Even a relatively low diversity college may be more diverse than the local neighborhood that the student grew up in, since most neighborhoods are highly segregated by SES (and SES outliers living there may try to “blend in” rather than advertise that they are much richer or poorer than most of the neighborhood), and many are segregated on other attributes like race, ethnicity, or religion, or the parents may bring the kids to activities that tend to attract people of similar race, ethnicity, or religion.