Let's take a look at the GamePad from Nintendo's perspective. Here's an ad that played around the time of the console's release:
From the ad we gain some important insights. First, we learn who the intended user is: you! The console is meant to be easy to use for men, women, the older, the younger, parents, siblings, and friends alike.
It seems Nintendo's goal is to create a personalized play experience that is different from that of every other player. The ad is shot in the first person, emphasizing that the wielder of the Gamepad holds a special power during the game. With a new private screen, "I can see you, but you can't see me" is one of the dynamics of play Nintendo is trying to introduce. In one scene, the player using the Gamepad pulls away from another player trying to look at his screen and says "No peeking!" This implies that the player using the pad has an enviable position.
I think overall Nintendo's desired user experience is giving one player access to a screen that no else has. This allowed that user to play special characters and have special roles. This, in turn, is supposed to bring the group of players together.