problem with it being a sport is that its very heavily dependent on the game, there is no universal game that people only play.
football everyone plays football. but in video games everyone plays a variety of different types of games all made by different companys.
I disagree, I don't think it has that much to do with the wider range of games available. Obviously some games have higher skill requirements than others, but to me all games have in common that you play the game with physical input. This can be with the mouse, keyboard, gamepad, wiimote, you name it. Point being, controlling your game is a physical activity which rewards good motor skills, speed, thought and strategy.
Obviously gaming isn't like boxing or MMA, but if you compare gaming to less physically intense sports like golf or darts you'll see a lot of parallels to gaming: they too reward coordination and precision, just in different ways. Hell, a lot of team based computer games have at least as much strategic depth to them as team sports. Only here your apm is on your keyboard, not the pitch and instead of kicking the ball you're moving and clicking your mouse.
tl;dr if you consider darts a sport, then so is gaming