On top of the weird naming (granted, it's an example, but going by Nintendo's rebranding in stores, they're moving away from the Wii & DS branding), I'm not sure if your route is the best way to go for a number of reasons (using an old controller, requiring a bottom screen, restricting 2 buttons from use by developers, etc.). How I imagined it was...
- The NX Handheld would be more like the Vita in its design with a more "Nintendo" approach. A single 540p capacitive touchscreen, clickable circle pads, a D-Pad, buttons, & 4 bumpers similar to the n3DS's (two of which are the click wheels that Nintendo patented, basically like a scroll wheel on a mouse). It can sync up with the NX Console & act as a controller (or plugged in to act as a wired controller) as well as enabling Off-TV Play.
- The NX Console's controller is about the same as the Wii U Pro Controller with a few exceptions. On top of being the main controller, it would have a 3.5mm headphone jack on the bottom of the controller. The clickable wheels would serve as the bumpers. There would be a touchpad to emulate the NX Handheld's touch functionality. And, like the competition, you can plug it in via USB to make it a wired controller.
Do I think Nintendo will go this way, I'm not sure. But I can hope, right?