@TheLadyFate Okay, so you've removed all the "Sims 3" entries using the Terminal command, and reset Origin, and Origin still thinks Sims 3 is ready to play? Yes, that's weird, and not easy to address. What you can try is doing a clean uninstall of Origin and installing it into a new admin user account on your Mac.
https://bluebellflora.com/how-to-uninstall-origin-on-a-mac/
If that doesn't help, can you download the base game on your other Mac? You can't install in Catalina, of course, but the question is whether the download itself fails. I don't remember whether it errors out at the end of the download or after the installer is supposed to launch.
Either way, if the new admin account doesn't work, try downloading the game onto your other Mac. Don't launch the installer, just look for it in Origin's Download cache. Open a Finder window, click Macintosh HD (or whatever you've renamed your hard disk, then Library > Application Support > Origin > DownloadCache. There should be a folder labeled with a bunch of numbers, and inside that, a .dmg file that's about 6 GB in size. You can copy this to an external, transfer it to your older Mac (the desktop is fine), and launch it by double-clicking.
You'll need your CD key to install the game, but you should be able to see that by opening your Origin game library, right-clicking on the Sims 3 icon, and selecting Game Properties.