About 42% of Harvard undergraduates get no financial aid, paying list price of about $63,000. Of the other 58%, the average financial aid is about $47,000, leaving an average net price of about $16,000. That may seem like “high need”, but if you play around with Harvard’s net price calculator, the parental income and wealth to get to that net price is in the upper range of the income and wealth distribution in the US (also note what levels are needed to get no financial aid). I.e. even the financial aid recipients skew toward the upper income.
In any case, it is not necessarily true that the family financial background of applicants is the same as that of admits or matriculants. The fact that the scions of wealth have many advantages in terms of achieving desirable college admission credentials means that they are likely to be overrepresented among the admits relative to the applicants.