What motivates professors to train Ph.D students who aren't nearly as smart as them in their schooldays?

As often happens, I'd disagree with significant parts of the implicit and explicit premises of the question.

For example, "as good as" traditionally makes most "sense" in the solitary-hero model of mathematics, in which there is some mystical gift/talent/genius without which one cannot reach the statospheric regions... blah-blah-blah. Further, "fast problem solving" is easy to see, but not much related to serious, long-term research enterprises, so to think about "solving problems faster" is misguided (I claim).

Yes, it is convenient for an individual (as it was to some degree for me years ago) to do well in contests, because this is an easy "success" to put on one's resume. In contrast, a natural affection for mathematics is hard to document, hard to "show", and so on. Many in my cohort at Princeton years ago had done very well on the Putnam, but turned out to not really like genuine mathematics (without quick gratification) enough to finish their PhD.

And, really, should one care in one's heart about the prestige of one's workplace? Sure, in some practical regards it may have advantages in pay and "status", but there are equally obvious downsides: pressure to generate status, as possibly opposed to genuinely advancing our collective understanding (for example). Tension and stress. Who needs it?

My own preference is for students who are genuinely interested in math, as opposed to viewing it as a career choice or a vehicle for ego. (E.g., some of my students who manifested a sort of mathematical "Oedipal" complex created needless troubles for themselves, and saddened me as well.)

To my mind, the operational difficulty in making interesting, genuine contributions to advancement of our collective understanding is not our "lack of heroic genius", but the commodification of research, and corporatization of universities, insofar as it leads to a fake objectification of "research". For example, success as measured by federal funding dollars. Srsly? The federal government as font of sage wisdom about human knowledge? "Even" NSF is driven by its accountability to congress, and certainly never having enough money to fund all the reasonably worthwhile projects people propose.

In summary, I like talking about interesting (to me) mathematics with people who are willing to not try to sucker-punch me or prove they're "better than me", but, instead, "focus on the math". Likewise, I have no interest in "proving I'm better than" anyone else, because I am less interested in those feral (if typical human...) interactions than in understand mathematics better.

The backstory on my possibly-extreme context for valuation of mathematics is that while I greatly enjoyed reading a variety of mathematics books in the public library as a kid, I had no idea at all that a person could make a living by doing such stuff, and was entirely prepared to just "think about math" as a hobby after 5:00 pm at some engineering job. The idea of teaching high school math was out of the question (crowd control), and community college was barely plausible (and pay not so good). The idea that one could teach "as little as 15 hours a week" and think about math the rest of the time and get paid pretty well was amazing and unfathomable to me at the time.

Plus, I claim that "math is not so hard", if portrayed less strictly-orthodox-ly than is common when "requirements" are used as a club to beat people with. That is, the ground question is interest and aesthetic, not "genius" (which is kinda a fake thing anyway).

(It is true that some people think their students should go off by themselves and come back only when they've done something amazing, but this seems to me a pathetic waste of the (supposed) accumulated knowledge/wisdom of more experienced people... This ultra-Spartan model connects to the bad hero-model, in my opinion.)

Wait, what was the question? :)

EDIT: ... and I should add that the main benefit of going to a fancy place for grad school was to get the idea that mathematics was not the dreary, grim, tedious, ugly thing that it may be portrayed by people who feel that they are inadequate, or don't really like it after all, but are "in too deep". That was a great relief!!!

So my question is, what motivates professors to train Ph.D students who aren't nearly as smart or excellent as them?

In most circumstances, research is not a competition. If I have a student who is working on a scientifically valuable project, then their work does not become less valuable or worthwhile if I have even greater accomplishments myself. Conversely, it doesn't become more valuable if I don't.

When I supervise grad students, I hope they will make the world a better place through research and teaching. How likely they are to do good work matters, as does how much time and effort I need to put into mentoring, but it doesn't matter whether they seem more or less promising than I did at the same age. I might make comparisons with other students to decide how to allocate my time (for example, we do this all the time in graduate admissions). However, if a reasonable investment in mentoring will enable someone to do good work, then I have no reason to care how they compare with me.