The one little loophole there that bugs me is the professional gamers who actually train to be able to operate that mouse and keyboard better, but as they are self described gamers they count themselves out I think.
Game and sport aren’t mutually exclusive, but professional gamers would say they’re in e-sports I reckon. And hand/eye coordination and dexterity is the same thing whether you’re clicking a million times a minute in DOTA, or ruining a walk playing golf.
I still reckon it’s the physical that makes something a legit sport, but computer gaming is physical, as is darts. (if darts isn’t a sport then neither is javelin chucking, or curling)