i know how you feel about the badges i don't bother with them anymore if i happen to unlock one as i'm playing that's fine by me
i'm just wondering with the uninstalling of the showtime one if there were left over files and registry keys that were missed per say (not by you but by the game ) that they weren't uninstalled properly and that the game "thinks" it's installed but it it isn't...have you tried the manually uninstall ? then reinstall
just in case you don't have the how to https://help.ea.com/article/manually-uninstalling-the-sims-3-showtime also if you have ccleaner run that and the registry cleaner part of it a couple of times to clean out the registry keys before reinstalling the showtime