Addicts can never admit that they have a problem
This is dead wrong. A lot of the time they won't, but there are numerous times where they do. In fact I know a lot more addicts who know they have a problem than addicts who don't, it's just difficult to stop
Addiction is mentally unhealthy, yes, but to blame video games for addiction and because of that, blame them for bad mental health, is silly. If parents allow their kids to play video games for too long at a young age then they're already proven to be bad parents. It's probably not the only aspect in which they aren't up to par and you can probably safely assume that there are other factors leading to bad mental health for video game addicted kids.
Also, violent people like violent video games, that's a pretty safe assumption to make. Why wouldn't they? But that doesn't mean that the video games
cause violence. Correlation =/= cause