But then I go into C:\Program Files x86 and there's an Electronic Arts folder with, whadyaknow, a The Sims 3 folder.. and inside? A Mods\Packages folder with a whole lotta crap in it. Other than a Mods folder there are folder called Caches, Game, GameData, Support, Thumbnails and a resource.cfg.
I'm going to go out on a limb here, and ask if you're running Vista x64. I had the same issue once I installed Vista 64; it installed to the Program Files x86 folder, and then Sims 3 didn't run properly. I ended up having to uninstall and reinstall (custom install, and tell it just Program Files, not Program Files x86) and now everything works properly.You may have to do that, although I would suggest that if you already have games saved, back them up to a completely different place before doing the uninstall.