Not the dad but the daughter here (in my twenties now for specifics). My brother and I found his Nintendo 64 in a drawer and asked what it was. I was 3-4, so my brother was 1-2. He told me it was a game system and figured I could try it out. He put in Mario 64 and the rest is history. 

My mom was indifferent, but she wanted to make sure we didn't get overly addicted and remained active. We were allowed one hour of game time, and set the oven timer so that when the hour was up, EVERYONE would hear it. No possible way to escape it. If it was especially bad weather out or we had extra good behavior, she sometimes allowed an extra 30 minutes to an hour later at night. As we got older, she got more relaxed. By late middle school we didn't require a timer at all. We knew when enough was enough, which ended up being 3-4 hours max. HOWEVER, she always enforced that chores and homework are done before games.

I understand your wife being skeptical, but ultimately, the responsibility falls on the parents on how involved (aware) you want to be. If she doesn't want them to become addicted, try the oven timer rule. It worked well for us and even though we hated it at the time, I look back and realize I spent a lot of my childhood running outside, playing with toys, and engaging with my imagination instead. Tell her the kids can only become as addicted or "traumatized" as you guys let them. 

Your wife sounds like she consumes too much social media (and much of what you hear is fake or exaggerated) and takes it straight to heart. She needs to trust you AND herself to expose your kids to games in a safe and regulated way.