That path (axioms to modern interests) is the path of history. But the path through which Math is understood by a person (the consumer of it) is not historic, rather it is through their mother tongue (whatever it may be) which as Chomsky tells us is a cyclic infinite formal system. Not acyclic.
That's all there is to it.
Because Math doesn't care to check back and "defragment", it is sticking to this acyclic system of axioms+logic which is simply—ironically in math's own terms—a local view on a large manifold (the formal system of truth), which is slightly incorrect at the fringes, the way a tangent space is not accurate away from the focus. And these inaccuracies manifest themselves into ever more tightly phrased problems of the Clay Math Institute.
They are Math language closing in on its own contradiction. Beautifully,
the process of abstracting a few very hard open problems is—surprise—Occam's razor in reverse (for physicists, this is “unification in reverse”).
I think the picture is simple and obvious.