[algeom]

This is actually not true in pure mathematics. Students usually do not publish with their advisors, there are no labs churning out publications, advisors do not apply their names to all the products of their groups, etc. In fact, the vast majority of math PhDs from almost all institutions end up leaving academia.

The small number of families seems more likely attributable to the small size of the enterprise and the difficulty of the subject matter. It's very hard to have a good idea and equally hard to publish it. I don't know of anyone in pure math who decides about the value of a piece of work based upon pedigree.