However, I wonder if there is something else going on here. What I noticed in the list (and in the article I am linking to) is that the early peak seems less obvious from a historiography perspective than it does from a contemporary perspective. So I can't help but wonder if there is a qualitative difference in the sorts of contributions that younger and older mathematicians make.

Thinking more in terms of fluid vs crystalized intelligence here, but wondering if there is something else. I know my fluid intelligence is not where it was ten years ago, but part of that is a greater understanding of the problem domain. I can see more possibilities, and I see different ones than I would have before.

So I don't think it is as simple as we might want to think but I do think that thinking patterns (and abilities) change qualitatively as we age in ways which are favorable for important contributions when viewed through the light of history. Whether a thesis review board would agree is a different matter.

It's almost certainly not true, at least among those that would be called mathematicians. This follows if for no other reason than the fact that it takes years to get up to speed with the forefront of any particular branch of mathematics, which is the prerequisite for testing out one's mathematical abilities (in the context of research). While you occasionally have a few teenage prodigies, most mathematicians won't get started on their career until much later.