Hello! I am not sure how much help I can be, but here are a few things you could try.
1.) Well, I noticed you said that 'the Sims 3 Supernatural' wasn't showing up in your 'My Games' section on Origin. Since there are so many Sims 3 games, Origin decided to pile them all together. So, if you hover over 'the Sims 3' in your 'My Games' section, and then press 'the Sims 3' button that has a little orange arrow next to it, it should take you to a screen with information about your game such as how many hours you have played it. If you scroll down on that page, you should be able to see all of the Expansion Packs and Stuff Packs for the Sims 3. The ones that are all bold should be the ones that you have. So, that should solve your problem of finding the game.
2.) If you go to your 'My Games' section, and right-click the Sims 3, there should be an option to 'repair' your game. Don't worry, I have done this before and it is not harmful to the game itself. If you have mods though, I would suggest moving the 'Mods' folder onto your desktop, repair it, then put it back in. It may take a little while but after it is done, try starting up the launcher and playing.
3.) Although it may say your game is up to date, it may not be. If you look below all of the Options on the left-hand side of the launcher where it says 'Version', you should have 1.69.43.024017 or at least, something 1.69_______
4.) Try reinstalling the Sims 3 Supernatural
5.) Go into the Sims 3 Game Launcher, press 'Expansions & Stuff' and then make sure all of your Expansion Packs and Stuff Packs are there and are checked.
Also, for the future, I would not recommend buying them Expansion Packs or Stuff Packs in digital copy. It's a lot safer and, in my opinion, easier to download/install.
I hope everything works out for you!
