Yeah, let's just try to manage that within what? Thirty seconds or less?
Also, I see no reason to explain what a game is to somebody. I've never seen a gaming commercial that does, Nintendo or otherwise. They see the product, a video game. That's as narrow as the category goes for non-gamers (the only ones who'd need to be explained to). After they know it's a video game, all you have to do is make it seem as cool as possible. This commercial does not do that, no, but your logic is flawed and would not only waste time, but bore people.
For example, the Zelda commercial could very easily just be a trailer of gameplay footage. That wouldn't explain what it is by any means, but it would certainly intrigue plenty of people.