As someone who does MMA, I don't really agree with the premise. Videogames don't make people violent, but neither do sports. MMA would on the surface be the most likely to cause violent tendencies, but it doesn't. Some people are more hot-headed than others, but that's as likely with someone who doesn't do any kind of sport. I agree with obsessive, but sports aren't bad for your health the way video games are. Yes you can get injured, and I've had some serious, nearly crippling injuries, but I'm in way better shape and physical condition than 99% of gamers, and normal people for that matter too. To put it into perspective, after 3 weeks sitting on my butt doing nothing, and not ever training for a marathon, I entered a 42km marathon on short notice to help a friend who was having difficulty with her training due to shin splints. I finished it, with muscle soreness and a couple blisters on my feet, but otherwise nothing wrong with me. That's not something people who don't do sports hardcore can do.
What your argument does do though is illustrate how ridiculous the argument against video games is though, since you applied the same logic that's applied to video games to sports, and it doesn't hold up in either case.