Or rather, it doesn't accurately capture income mobility in the same way. It compares the economic mobility of equally selective private universities to that of BYU, and shows that BYU produces much less income mobility.
The first flaw is that the reason those other universities are selective is largely on the basis of grades and scholarships and economic opportunities. More people want to get into Duke or Stanford because of the perceived (and real) networking effect that attending those schools will have on the trajectory of their careers. BYU's incoming students want to get in largely for cultural and religious reasons. So the relative selectivity doesn't capture the same things.
Second, its flawed because in many ways, BYU operates more like a public university, with a premier flagship and smaller satellite campuses that don't have the same academic thresholds, and which are perceived to be less than the flagship in various categories. There is no equivalent of Duke-Wilmington or Stanford-Temecula as there is with BYU and BYU-Idaho.
All of that said, I do think BYU could do more to reach out and accept economically marginal students in some cases. But the problem isn't nearly the issue that Brookings made it appear.