The dirty secret is that they haven't really had those supposed educational advantages for a long time, if ever.
I attended undergrad at a large university that was (and still is) around the 75-100 range in US News rankings. I then attended graduate school at a top-10 US News school. My expectation coming in was that compared to my peers in undergrad, the undergrad students I was teaching as grad student would on average be smarter and work harder, and would have better resources available to them. All three were false. They were on average more ambitious and more demanding (in terms of e.g. asking for points back on assignments), but not smarter.
There are very real advantages to attending an elite school, but they are primarily social rather than educational. Having letters of recommendation from top scholars helps you get into grad school. Making friends with other people attending an elite school can provide all sorts of opportunities down the road. Just having the name on your resume will open doors for you. The undergrad education itself, though, probably isn't better.