You have to differentiate between what the "Wii U" does, and what the "Virtual Wii Mode" does, because as far as the console goes, they're two different things. You can't expect that a new console to increase the resolution of everything related to the system it's retrocompatible with as if it was pushing a button. It would require a lot of work, because forcing standard definition assets, such as textures and HUD elements, would look TERRIBLE, so you would need to redo them on higher resolutions, besides all that other work that goes into tweaking everything so it looks presentable, or even works in a stable fashion.
When the Wii U goes into vWii mode, stops being an HD console and just becomes an imitation of a Wii. The vWii works at 480P because it's the resolution at which the original Wii worked. What the Wii U side can do doesn't matter. Just like PS1 games on PS2 have their original 240 resolution even if the PS2 is (or attempts to be) 480, or GBA games not using the entire screen of a NDS.
It's not like, let's say, the Xbox One's "Retrocompatibility". It's actually less that and more reworking old games to work with the newer hardware. That's why they have improved resolution, visuals and performance most of the time. That's pretty cool, and that's why they also ask for money for it.
Of course, there would be no reason to set the Wii U side to 480P having better resolutions, that it's true. But you have no choice under vWii. It's a "system within a system", as I said.
Also, I remember reading that Wii U had a better color range (or at least was more colorful) with Component cables than HDMI, but I've never tested it myself, so HDMI isn't the "only" good choice. I think Xbox 360 was the same.
No, WiiU has the Wii's video socket too (supporting all Wii video options, plus D3/D4/D5 (720/1080) over YPbPr cables)
(There also are plenty of VGA or less HDMI or DVI displays, though I doubt any design consideration has been explicitly given to those)
Even 240P? I thought it couldn't do that, but maybe that's only on the Wii U side of it.