The athletes who competed on our chess teams were as physically and emotionally exhausted after competitions as any of my traditional sports athletes. So, yes, chess is a sport.
Note that 5-minute blitz games definitely require some hand-eye coordination and timing, using the same hand to move a piece and hit the clock and it can get a lot more hectic than golf and bowling when someone with less than 20 seconds on the clock is trying to execute a mate.
being exhausted itself doesn't qualify. if so, every college student cramming for exams would be labelled athletic.
the hand-eye coordination you wrote of is laughable! your example simply doesn't compare to hitting a baseball or golf ball. or, any other sport. it reminded me of when i was a kid, trying to eat dinner quickly to go play actual sports.
now, that's hand-eye coordination.
Still a good summary of the issues, in the link below.
http://londonchessconference.com/a-question-of-sport/
To wit -- "[Chess] is not recognised as a sport in the UK and receives no public funding. It is worth reminding ourselves why the International Olympic Committee and over 100 countries recognise chess as a sport."
the article is not a good summary. none of the ten reasons mentioned can really differentiate chess from any other board game, aside from it's history.