I'm a fellow gamer, have been since I was 6 years old. I've played shooters when I was 8 (NES), violent bloody 3D shooters like Duke or Quake when I was 10, mature-language, violent and also bloody fallout when I was 12, and hundreds of various pegi-18 games after those. I'm not a murderer, I'm not particularily violent, I've never tortured an animal, etc etc. In general - I think I turned out fine.
In spite of all that, I'm going to pay attention to what my children play and how they do it. In case there's a hint of excess aggression or any other game-inspired negative behavior.
So definitely supervise. Try to play with your kid and try to "point" him in the right direction, which is working on improving your skill in a particular game and not reacting negatively.
Whenever you feel it is necessary, react. And moderate. It's your right to do so and it's better to be safe than sorry.
If by moderated you mean limited in duration, then I think that it should be done. Mainly because small children will spend all of their time playing, completely neglecting any chores, homework, playing outside with friends - virtually neglecting everything else. Some people let children play only on weekends.
personal thoughts, no real scientifical justification:
I'm a strong supporter of a strong distinction between computer game world and a real world. Ten or twenty years back the distinction was pretty strong with "bad" graphics and all, but now the border between real and game world is getting blurrier and blurrier. Make sure your child plays games which are clearly "not true". The world has to be totally different, sci-fi, fantasy, cartoon-like - so that there's no confusion between reality and virtual reality. I will not permit my kids to play games like GTA, Call of Duty or Battlefield simply because they look much too realistic. Where there's a strong unrealistic setting (Quake, Skyrim, Jedi series) there's no confusion and I think these games are safer to play.