This article is quite mistaken, at least from the perspective of an academic mathematician. (Maybe it is more accurate in the humanities.)
(1) This year, two of the three candidates whom we interviewed got their Ph.D.'s at non-top-20 universities. We made offers to both of them. One of them turned us down, in favor of a better offer from another R1 university.
The third got his Ph.D. at a top-20, but not top-10 university.
The last person we hired, in a previous year's search, also got his Ph.D. at a non-top-20 university.
(2) My own university is not in the top 50. Most of our own Ph.D. graduates, if I limit to the strongest and hardworking ones, have been successful in getting either academic jobs at good universities, or very appealing non-academic jobs.
(3) In mathematics at least, the top universities offer a much better graduate education than non-top universities. For one, the absolute leading researchers are at the top universities, and there is a big gap between these people and merely very good researchers. (I say that, counting myself in the latter category.)
But still more important than that is the quality of the peer group. In grad school (at least in math) you learn much more from your peers than from your professors, and Harvard, Princeton, Stanford, etc. attract the most talented, ambitious, and hard-working students. That is why these places are so good.
(4) "Why don’t prospective graduate students simply limit their applications to favored elite institutions? The answer is often financial, and, again, speaks to privilege and discrimination endemic to academic culture. The most prestigious universities – the Ivy League, University of Chicago, Stanford University, the University of California system – tend to lie in the most expensive parts of the country."
Well, for one, these places all pay higher stipends. I don't know of any grad students in math at any of these places who have had difficulties paying their living expenses, except for those who were simultaneously trying to raise families.
Ambitious students who can get into elite universities, usually should. And exceptions are typically in the case of elite research groups in otherwise non-elite universities.
I am not saying that there is no privilege and discrimination in mathematics departments (the gender ratio is notoriously bad), but I find the author's criticisms off-base.