@Phaewalker You don't need to subscribe to EA Play or otherwise hand over any money to play Sims 3. I have a Steam install, and my accounts are NOT linked, and the game runs fine.
Please uninstall Sims 3 using Revo Uninstaller (the free version is fine). Remove the EA App with Revo as well. Restart your computer, reinstall Sims 3 through Steam, and don't install the EA App at all yet. Try to launch the game through Steam and let me know whether it works. If it does, you can create a desktop shortcut for Sims3LauncherW.exe and launch the game using that. Or you can use TS3W.exe and bypass the launcher entirely.
If you still can't play, reinstall the EA App, and check your game library to see whether Sims 3 is available to download there. If it is, uninstall the Steam version with Revo again and reinstall through the EA App, then see whether the game works.
If this doesn't help either, or you'd like to find out whether something else is blocking the Steam version from starting, please run a dxdiag and attach it to your reply.
https://help.ea.com/en-us/help/pc/how-to-gather-dxdiag-information/