How big of an advantage do you have if you can pay full tuition?

So, the bottom line is that applicants to the most exclusive schools are overwhelmingly no-need, or low-need, due to the nature of the selection process itself. The best-scoring, highest-achieving students will be skewed wealthier. This does make sense to me.

However, when you hear of 30,000 - 40,000+ students applying to each of these schools, it just is not possible that the majority of these students (even if they are high scoring and high achieving) can afford a quarter of a million dollars (and up) for undergraduate education.

Are there really that many applicants whose families can afford this level of tuition?