I question whether mathematics is really as holistic and intertwined as some people are making it out to be.
Certainly there are a few freaks of nature out there who can understand 20% of the mathematics out there and incorporate ideas from 10 different subfields into their work. A larger group of us are capable of getting the big picture though maybe not all the specifics of 4 or 5 different subfields, at least to the extent that we know when to reach out to an expert. Many graduate students, and most of them once one leaves the world of the top ten or twenty departments, are just capable of learning one subfield well enough to write a dissertation narrowly focused on one problem in that subfield, ignoring all the wider connections if indeed there are any. Most published papers are written by people who have never done serious work outside a single narrow subfield in their entire career, even if the same is not true for the best papers.
A professor or a department may choose to aim its education at the future Fields Medalist (or, somewhat more broadly, the future NSF-or-equivalent-research-grant-receipients), but is this really fair for the other 19 students in the room?