I would say that olympiads build some, but far from all, of the skills needed to excel at mathematical research. I'd compare it to running 100 meters versus playing soccer. Usain Bolt is probably a better soccer player than the vast majority of the population, because he could outsprint anyone and because he's generally in fantastic shape. But that doesn't mean he's going to be able to play on a top team.
Being a successful researcher requires the ability to learn new fields of mathematics, and develop ways of thinking about them that others haven't. It requires the discipline to spend months or years returning to a problem and trying new angles on it. It requires, or at least is strongly aided by, the ability to communicate and "sell" one's results, in writing and in talks. It requires the ability to write good definitions, that will be useful and cover the boundary cases correctly. It requires the ability to form an intelligent guess as to which unproven statements are true and which are false. It requires the ability to hold a complex argument in one's head and play with it. It requires, or at least is strongly aided by, the ability to find clever technical arguments.
I would say that olympiads are very helpful in developing the last skill, somewhat helpful in developing the fifth and sixth, and not at all in developing the first four.
I definitely, at some points in my research, find myself needing lemmatta which would be fair to put on an IMO or a Putnam exam. And when that I happens I feel myself relaxing, because I know I can do that. But I also spend a lot of my time trying to learn how to think about a subject, or figuring out what to prove, or trying to figure out how broadly a phenomenon holds. And those are not skills which I found olympiad training helpful in.
For the record, I was very competitive in my High School years, but not a world level competitor; I was the first alternate to the US team in 1998 and, during my senior year, I regularly came in somewhere in the top 10 spots in national contests. I've heard similar views expressed by Kiran Kedlaya and Ravi Vakil, who were much more impressive competitors.