Of course, intelligent people are drawn to "deep" problems. Being willing and able to solve "deep" problems is practically the definition of academic intelligence. Actually, most people like "deep" problems more than shallow ones, but only intelligent people have the mastery of the basic techniques required to takle the deeper problems.
I'll admit this is all very much [citation needed] stuff, though. I'm not certain.
Citation? Wiles proved FLT at age 40. Perelman proved Poincare at age 39. Richard Hamilton was 50 at the time.
But even with that great mathematicians START doing deep problems early in life. By the time their 25 they've been doing deep problems for a decade at least. Terrence Tao, for example, was doing calculus at age 7. He got a gold at IMO at age 13. These great mathematicians like Tao, Goedel, Neuman have long gone beyond grill and drill math before age 20. So the notion that they just recently stopped solving shallow questions is absurd.
This is a ridiculous myth. As a truism, person's creative intellectual output is greatest when they are working hardest and have the greatest number of tools of creation available to them. I could go down a long list of mathematicians (famous and not famous) whose most influential work was done later in their careers.
As you have defined the terms here, what you say makes no sense to me. It seems to imply as a direct corollary that solving deep problems does not make you any better at deep problems, whereas solving shallow problems does make you better at deep problems. I can believe the last half so long as it is only a supplementary form of training (a crude analogy might be of a virtuoso musician practicing scales), but the first statement beggars belief and is contrary to all my experience.