I read somewhere recently that the purpose of prestige/Ivy League schools is to provide the very wealthy and the very talented/ambitious the opportunity to mix, where you won’t be able to tell the difference.
That has mainly been the purpose of the wealthiest schools. Elite sends their kids to the same schools, to be prepped for a role in society similar to the one their parents have/had. They're finishing schools essentially.
Then there are schools like Rose Hulman or Curtis Institute of Music that specialize hard. They're very prestigious, but you also have to know about them to even think about going to them.