@MYOS1634 , not Amherst or Williams that I’ve ever heard about. UMich grads show up once in a while because they have such a strong math program generally.
@ucbalumnus , I would not include Berkeley in the handful of top-5 programs for this purpose only because it is not a private U and has significantly less money to lure students. (Of course it is a highly-ranked program by any other measure, including all of those that are more relevant than how much they can pay their Ph.D. students.) The schools I would definitely include are Stanford, MIT, Harvard, Princeton, maybe Chicago and Brown. There are huge discrepancies in the amount of money Berkeley and UCLA can throw at students versus what Stanford and Harvard can. Run through the current list of Ph.D. students at Princeton, MIT or Harvard, and you will find very few Big State U or small-LAC undergrads.
That said, there are plenty of prestigious -enough math programs that are willing to admit outstanding students from a wider array of undergraduate institutions, and even the tippy-top private Us will let in a member of the great academically unwashed once in a while. But my point is that the broad generalizations in the posted article are so broad as to be absurd.