New Super Mario Bros. is a game that appeals to children and adults. (now, i don't own a copy of any of the japanese versions of these games, but i'm sure they have the same feature.)
You don't have to take the invincibility power-ups (as far as I know). If you want to keep trying to beat something that you're clearly struggling with (you died 3 times in a row) the way it was intended, then simply don't take it. For a child, some puzzles or bosses or stages seem insurmountable, and odds are, they don't particularly care about beating the game the 100% honest way.
This is drastically different compared to games in the NES days, which would downscale the entire game's difficulty, or remove a harder difficulty option altogether (which makes so little sense that I can't believe who the hell thought it was a good idea to ever do that to a game).
that said, the only game that i know you don't have to take the invincibility is in Super Mario 3D Land; and i'm going to assume it's the same for every other game.
I think this recurring argument is an over-exaggeration of an issue that doesn't exist anymore.