Hi simmum.
Whenever you register your hard disk on the Sims 3 site, you are then able to download it using Origin. This applies to the base game, as well as all of the expansion and stuff packs. As for uninstalling all of the expansions and stuff packs, you could try doing just the base game and see if it will work. If it does, then you're good to go. If not, you haven't wasted much time with just the one re-install. In this situation, I don't know if there's a way around the uninstalling malarky.