I know plenty of very good mathematicians who did not participate seriously in this kind of thing. (Disclosure: I work as a mathematician at a research university.) I'm not sure why the only ones quoted here are mainly ex-winners. Indeed, while these competitions may have the positive effect of putting like minds together, it's possible that they have the negative effect of discouraging those w/o the aptitude for this particular kind of competitive sport (which is what it is), or who do not have access to the kind of coaching and practice that successful competitors often have.
The one math genius I know who despised Olympiads, ended up leaving academia over a famous but wrong proof.