I should have posted this awhile ago, but better late then never.
I did indeed get my Wii back, freshly repaired from Nintendo. Actually, they sent me a different Wii altogether. Fine by me, especially since they managed to keep all of my data intact. All I had to do was redownload my VC games (free, obviously). The only casualty is that my Miis aren't editable anymore, but that's hardly a big deal.
I was surprised to find that even my VC save states were intact, depite having to redownload the games. Pretty sweet, since I had progressed pretty far into some of them.
All in all, it took Nintendo a little less then 2 weeks to get my Wii and send it back. A good experience, well, as good as having a bad console can be, I guess.
The one good thing about waiting before making this post: My new Wii isn't showing any signs of graphics glitching or overheating.
-Rob