Here in Australia college really is meritocratic. You need good scores to get into good schools, and all the schools are free (you have to pay them back after you graduate, but only if you earn enough money. The point is if you are accepted to a good school you can afford to go). So what's a wealthy family to do? Private high school. Many of which charge over $12,000 a year.
Elite colleges don't have better instruction, they have better research projects. Go to an elite college and your Teaching Assistant is doing Nobel prize work as his day job. That has obvious advantages, but the point is that the advantage isn't so much from teaching skill as other factors. Factors that don't apply to high schools. The education isn't that much better at a private high school than at a public one.
What does apply, though, is sending your children to school with the children of the rich. Not just because all of the kids there are studying all of the time (so your kid does it through peer pressure) but because the connections they make will keep them in the aristocratic class more than any education they get.
“Our new multiracial, gender-neutral meritocracy has figured out a way to make itself hereditary.” - The Birth of the New American Aristocracy
And in one of the enduring absurdities that always makes me crazy:
You see, when educated people with excellent credentials band together to advance their collective interest, it’s all part of serving the public good by ensuring a high quality of service, establishing fair working conditions, and giving merit its due. That’s why we do it through “associations,” and with the assistance of fellow professionals wearing white shoes. When working-class people do it—through unions—it’s a violation of the sacred principles of the free market. It’s thuggish and anti-modern. Imagine if workers hired consultants and “compensation committees,” consisting of their peers at other companies, to recommend how much they should be paid. The result would be—well, we know what it would be, because that’s what CEOs do.