The pitch behind the GamePad experience appears to have been roughly: "Do you like game controllers, but wish they were the size of a large breadbox? What if this breadbox was only usable for two hours at a time and had to be played adjacent to a power outlet?"
Hey, really well written reply! For brevity i'm only quoting part of it. There were leaks/rumors during the gamecube days that Nintendo liked the idea of a screen in the controller. They did asymmetric gameplay with the GBA and Gamecube during the day and I'm sure the dual screen idea also inspired the DS. They loved the idea!
But gameplay wise, Nintendo wasn't able to deliver on the gamepad screen. In fact the BEST uses of it were from third parties. COD Black Ops 2 allowed two local players, one on the TV and one on the gamepad. Rayman Legends was great asymmetric platforming. My wife is not a gamer but we enjoyed that tremendously, i'd do the platforming on the TV she'd do the cut the ropes and puzzles on the gamepad.
As another example, think back to the Wii. Motion controls were this gaming dream! Right from the start people imagined 1:1 accuracy sword fights, and light sabers, and throwing spare ammo clips to your team mates in FPS. Those were mostly false hopes...but Nintendo DID deliver on the concept in skyward sword. It was their way to say "hey this is what we always envisioned and here it is, working, and good". Wii U, unfortunately didn't have that game. And that is actually shocking for Nintendo who frequently designs their hardware around a certain game they have in mind.
That said, to the OP's point the Wii U didn't get a fair shot. I traveled with mine through airport security recently (to play zelda where I was going) and the airport agents had never seen or heard of it. It got no attention. And it wasn't for lack of games, I was super impressed with Splatoon won shooter of the year over Destiny, COD, Halo, and Battlefield! And to your point, people are jazzed up the upcoming Switch games which are remasters of excellent Wii U games (that I guess most folks didn't get to play). The problem was that no one, not sony, microsoft, or Nintendo were prepared for consumers that were THIS spec hungry.
PS4 sold well out of the gates (despite having no games) because of the specs. Games like The Order 1884 or the equivalent of playing specs. Sony and Microsoft have re-launched their systems with higher specs because that is all people want.
So my theory with the Wii U is Nintendo predicted the power levels of PS4 and XBone and crafted a machine with a similar GPU centric design, and feature parity (DX 11 equivalent). They got burned with the Wii which didn't even have programmable shaders! If PS4 and XBone were doing 1080p, then Wii U should be able to get a port at 720p and some reduced effects or frame rate. Next, they wanted a touch screen in case mobile iOS/Android devs wanted to port to Wii U. And lastly they wanted Wii backwards compatibility so they could pitch the "upgrade to Wii U" campaign. If only 10% of Wii's install base fell for it they'd have huge numbers out of the gate that couldn't be ignored by third parties.