Dexterity is important in speed chess, whether played OTB or online. Too much speed chess online can leave you with a sore wrist and shoulder. I’ve been suffering the past few weeks, and yet I cannot stop playing. Maybe I need to check into rehab.
I know a lot of people who hold the view that Chess is an excellent means of training the mind in logic and shrewd calculation, prevision, and caution. But I don't find these qualities reflected in the lives of Chess Players. They are just as fallible, and as foolish if you like, as other folk who don't know a Rook from a Pawn. But even if it were a form of mental discipline—which I take leave to doubt—I should still object to it on the ground of its fatal fascination. Chess is a kind of mental alcohol. It inebriates the man who plays it constantly. He lives in a chess atmosphere, and his dreams are of gambits and end games. I have known many an able man ruined by Chess. The game has charmed him, and as a consequence he has given up everything to the charmer. No; unless a man has supreme self-control it is better that he should not learn to play Chess.
J.H. Blackburne
http://chessskill.blogspot.com/2009/01/blitz-addiction.html
Because he's better. Which has nothing to do with why chess isn't a sport. He moves his pieces because his brain is better, not because his hands are better.
Hand movements that become very precise, improved dexterity, improved reaction times, and improved arm speed in tight time situations, these are not physical skills to you?
They are. All of which have nothing to do with someones ability to play chess. Or be better at chess. The results of a chess game aren't decided by who has the best dexterity. They are decide by who has the best knowledge of chess.