there is some serious hand dexterity and hand eye coordination going on too with most of the competitive games. The mental puzzle games are probably not the ones that most competition is aimed at.
Ultimately, I agree. I feel that gaming is a sport. Watching a world champion StarCraft player manage 6 bases with hotkeys is something to behold and not something an average mortal brain and hands can find success with. I'm just drawing the line in the sand before reflexes are considered. Considering reflexes we may need to consider musicians as athletes as well. But the OP asked if gaming competitions are sports, assuming in the light of football, baseball, basketball etc., and I feel a Chess application would apply better. I find College teams popping up and they are treating them more as clubs instead of athletic teams. I think the industry's nailed it with the sub-group E-Sports, neither here nor there.