They didnt say they believed they would recapture it, they said "some" of it.
And yep they did say "they are addressing the family market".
Nintendo has always been the most kid friendly console out there.
Not necessarily, when the Gamecube came out, my younger siblings had one, but they definitely got much more use out of the PS2.
While Gamecube DID market nearly exclusively to kids, there was definitely a lot more on PS2 for children to enjoy; for starters, the PS2 had DVD playback, so kids got to enjoy their first DVD player with their console. Second, there were better games, such as Katamari, Guitar Hero, Singstar, and Okami. Just because the PS2 had the additional benefit that it marketed strongly to other demographics as well, doesn't mean it wasn't better than the GameCube for kids too.
I think Nintendo went a little too far with the handle so little kids could carry around their Gamecube everywhere like a lunch box, or the big green button, or the giant L & R triggers with the toy-like springy noises included. I know that even my younger siblings were a little embarassed to have the "kiddy console" during the GameCube era. So I don't think most kids liked the GameCube better than PS2 either.
I describe myself as a little dose of toxic masculinity.