I've studied mathematics at the graduate level, and I agree 100% with your assessment. I found all the criticisms in the article to be fair and accurate. I'd like to provide a little more perspective here.
I think people don't understand how broad a field mathematics itself is. In graduate school, one will gain a broad knowledge of one particular subfield (e.g. algebra, analysis, topology/geometry, or discrete math, for instance), a very deep knowledge of a particular subject within that subfield (e.g. sheaves of smooth functions on manifolds), and become the world's leading expert on a particular subject (the dissertation). Along the way, one will certainly pick up at least a couple of courses in perhaps 2 or 3 other major subfields sufficient to pass the comprehensive exams set by the department, but that's it.
So, a mathematician graduates knowing a lot about very little compared to the whole of mathematics. Over the course of an entire career, one will usually pick up bits and pieces that were neglected in grad school, and perhaps enough knowledge to contribute in maybe one or two additional subfields.
This, of course, is how it works for an average mathematician -- the average Joe or Jane who makes it through grad school. Superior mathematicians might contribute more broadly (Erdos comes to mind), but they're sufficiently rare that we can practically discount them. Charlie Epps goes beyond this because dramatic license allows (indeed, practically requires) him to.