Here's a scenario to try shed some light on what you mean by "well-rounded" for the rest of us and for yourself.
My friend and I were both in the top math class at the end of high school. He is a hard worker, and I am a natural talent. Whenever we were taught something new in class, I would go away thinking that I understood what I was just showed and I could apply it with a high degree of accuracy to whatever applications tests might throw at me. My friend went away and studied the topic to death, learning from various textbooks to get a feel for all possible questions he could ever be asked.
Now since I had much more time to fondle with while he studied profusely on these topics, I would indulge myself in all kinds of topics that weren't in my curriculum in school (mainly by reading up on problems that users here on PF couldn't solve in the homework help section).
Anyway, much later down the track when we were revising, I would slowly but surely get an answer to that topic covered long ago. My friend would rush through it using short cuts and tricks along the way and get the answer much faster than I could.
In the end of it all, he came first and I came second.
Now a question to you. If it were only my friend and I that were the two candidates for this search of yours to find the "strongest well-rounded mathematician", who would take the cake?
If you chose me then isn't it possible that given more time my friend would also learn everything I have learned and to top it off, he would be better at these topics. He would've been exposed to more proofs, more examples, more tricks etc. and he too would most likely develop a more natural affiliation to understand mathematics with time. Plus, at the present moment he would pound me in a competition on the topics that we have both learnt.
If you chose my friend, then we could argue just the opposite and it would obviously look foolish to choose him and then I come along and could give some insight into a problem that he has no clue about.
And then there's the issue that for PhD's and beyond, they usually follow their own paths that they prefer, specializing in some field of mathematics. How do you compare them then...?
of course that entails a lot of material, but I'm arguing that those four things are the minimum required in order to have a broad mastery in problem solving of existing and new problems.
