I think you said it right. It's fine for children of any age to play video games, as long as it's appropriate for the child's age. I don't think you would want your 6 year old playing GTA of Call Of Duty. Or even something that seems cartoony like Conker's Bad Fur Day. Which I might add started out as a child's game until the developers got perverted.
But pretty much any Mario, Zelda, or Sonic game would be fine for pretty much any age. Then there's games that are aimed for more of a younger age bracket like Banjo Kazooie. I'll admit, I'm 37 and LOVE Banjo Kazooie. But hell I still watch Fraggle Rock, so go figure.