It helps to understand Nintendo's thought process when they were designing the Wii U. Originally the GamePad was a second screen that sat by the TV. In their mind, it's an inexorable second screen that might as well be a part of the console. Wrapping a controller around it makes people think "oh just drop the screen and give me the controller" but in Nintendo's mind the screen is as fundamental a part of the console as the video-out ports.
Obviously it's inconvenient if your pad is in some way broken, but the "intended use case" for the Wii U is that you interact with the GamePad to boot the system, often via the quick launch menu, and to do system-level things like settings and shopping. Only when you boot a game that utilizes an alternative controller do they expect you to sit the pad back in its dock, and pick up that other controller.
There are plenty of annoying things here if you try to force the Wii U to operate like a different machine, but that's what Nintendo does these days. Their hardware is gonna be different to use somehow, and if you try to cram it into old (yet perhaps still reasonable) paradigms you're going to suffer.
Really well put post. One of the main reasons I went for Wii U was for something different, and whilst many are trying to compare it with 360/one/ps3/4, you're right, it's a different kettle of fish altogether.
I see the gamepad as part of the OS, and part of the console. I don't see it as a kinnect type thing they can just remove.
I cannot stress enough how off-screen play has improved my gaming experience, it has made co-habiting with my girlfriend a very smooth transition from bachelor life, (going from a two tv living room to one!), I actually think I have more time for gaming these days, now that I don't need a tv for it!
I don't see the big deal with grabbing the gamepad and the pro controller - normally, the gamepad sits in it's charging cradle when not in use, with the pro beside it on the coffee table. If I'm having a gaming session, I'll just grab both to have in front of me, I'll start with using the gamepad to boot up - change tv input (if I'm using the tv), quick check of eshop/friends/miiverse, and if I have any eshop codes to enter I'll do that, then I'll grab the pro if it's a game that feels better with it.
I try to avoid using credit card to buy eshop credit (wish I could use paypal or other bank card), but if I was taking the time to enter that number, I'd have time to use the gamepad to do it, no big deal.
If the gamepad is broken, I'd get it fixed, as I'd have to get the xbox or ps fixed if a part of that system broke. If it was just the cable broken - I'd replace the cable.
I love the pro controller, but I don't expect to be using it for anything other than gaming. The system functions, eshopping, etc. would be pretty clunky without the gamepad, the way those things work, the information and button layout, they just work better with the pad screen.
I get the issue of the OP, and I don't think it would be crazy to allow those functions to be done without the gamepad, but the user interaction in doing those things would be a lesser experience, and probably something the developers don't want to cultivate. For example, using the gamepad to navigate the eshop is great and I like the interface of the eshop a lot, using the pro controller to navigate it, is a bad experience, it'd need a complete redesign to be easily useable with it, and the eshop would end up something like ps or ms storefronts.