Interesting query. Chess is an activity that requires an immense level of concentration, stamina, and skill to play at a high level. Hallmarks of a sport.
But chess will certainly not "get you into shape," and I don't think anyone would consider chess an exercise, whether or not it is a sport.
I'd put it on about the same athletic level as auto-racing. It looks easy and leisurely enough to the casual viewer, but in reality it is an intense struggle where both sides can find themselves exhausted physically at the end of it.
Cricket is a sport but I would hesitate to consider the players necessarily athletic.
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You don't consider cricketers to be athletic?
You don't consider bowlers who bowl 90 and 100 mph athletic?
You don't consider wicket keepers who litterally throw their bodies left and right of the stumps, catching the ball with one hand athletic?
You don't consider slip fielders or fielders in the outfield who chase after the balls, and deliver spot on return catches or stumps athletic? Who btw catches the ball bare handed not gloved like base ball.
You don't consider batsmen who run miles between the wickets, and hooks and drives the ball athlethic?
Sorry sir, but clearly you have not seen much cricket...there is no position in cricket that does not require an athlete.