@xboxbabe70 I split your comment into its own thread, because the other thread is several years old, and things have changed since then.
It sounds like Pets is showing exactly as it should in Origin—it no longer gets its own entry in your game library. But the fact that it's not listed in the launcher is of course an issue. Occasionally, the issue clears up on its own, perhaps after rebooting the computer a couple of times. But since you're likely past that, you can try clearing Origin's cache, and if that doesn't work, uninstalling and reinstalling Origin "the hard way." This wouldn't affect your TS3 install.
https://help.ea.com/en-us/help/faq/clear-cache-to-fix-problems-with-your-games/
https://help.ea.com/en-us/help/origin/origin/manually-uninstalling-origin/
If these steps don't help, the next recommendation is to contact EA customer support, as your code may have been deactivated. Sometimes this is a mistake; other times, it's due to EA belatedly identifying a bad code, usually bought from an unauthorized third-party seller. The support rep you talked to should have been able to tell you whether this was what happened in your case, but either that didn't happen, or your code was fine. (I wonder if the code you were provided was your original one, which is why you couldn't register it a second time.)
One more thing you can do is uninstall the game and clean your computer's registry with a tool like Piriform's CCleaner (free download). Then redownloading might help. If it doesn't, though, please post back, and I'll forward your information to someone from EA. Hopefully then you'll get a better answer, though it might take a couple of days. You're absolutely right that this shouldn't be nearly this difficult to resolve.