Well, I am not going to argue with his Holiness

In general I agree with his thesis. However, he is IMHO making a big assumption which which I do not concur. My experience has been that just because someone has a human body, doesn't make them a human being. OK question of definition and degree perhaps. In addition, in making the comparison with other animals ie that "we all want peace and security," he is eliding a fundamental fact of living on this planet, the fact that most species need to eat each other to survive. Hungry fish are not about to grant each other peace and security. Mr Zebra is not asking Mr Lion to eat grass. Mosquitoes will not leave me alone. That is the - literally - the 'nature' of this 'reality'. (It is not of course the only reality - I am loath on this forum to elaborate).
The human being shares many characteristics with computers (it is interesting to ponder the fact that computers may indeed be made 'in our image'). Computers have a 'body' and a 'brain' which runs programs. The brain has levels too, and there are fundamental programs eg aggression and hunger. In a computer such programs are called firmware (loosely - for any pedants who may be stressing). In a human we call them instincts. The extension then is that the personality is just this, a program. Who is the programmer ? This is more firmware: the theory is that the brain programs itself, based on experience - and guess what - we are developing computers which do just this. Consider also that we know that the brain can be re-programmed (sometimes called brainwashing), just as we know brain damage (indeed more basic physical damage) can cause personality change.
This does not mean that humans are not evolving to higher levels of consciousness. Some are, most are static and others are regressing. Logically (as it happens), it depends where you are starting from and the amount of effort (practice) you put in, where you get to. I guess that might be what is worrying you about these violent games. For me it seems most likely that someone who is younger (less experienced/programmed) or who is unstable/borderline psychopathic might be more affected by than a person who is more integrated/evolved. But it also depends on what other programming is going on. Some one growing up in Afghanistan is getting a whole different set of programs than someone growing up in the UK. I would agree with you also that someone like the Dalai Lama would be likely to consider such games to be counter to his desire to see more love and compassion in the world.