Math and chess are two different things. It is not a rule that anybody good at math should also be good at chess. Yes, we agree that chess involves strategic skills and is a calculated game, but not all mathematicians are up for it. Mathematicians and chess players do share some unique qualities together but that is because the two things require a similar skill set.
However, the employment of these skills is different which is why the two can never be together. Yes, maybe it is easier for a mathematician to learn chess but he will still need to learn the game, like every other person.
I would have to dispute the fact that all mathematicians are good at chess. While chess is a game that requires strong strategic skills, being a good mathematician does not necessarily mean that they are going to be real good at chess. Being good at chess means that you have to constantly play just like you would when it comes to playing and becoming good at checkers. Reasoning and strategy are certainly skills that any great mathematician has.
I would have to say though in order for them to become great at chess they would have to play the game and be exposed to it. Having those computational and reasoning skills would help but it certainly does not mean that they are going to be great at chess. I would most certainly say that a more experienced chess player would be able to defeat a strong mathematician who has little experience or has never played chess before.