Entire game shows as not installed

@Branwen84  Try running an App Recovery, which you can do even when the EA App won't open:

https://help.ea.com/en-us/help/faq/clear-cache-to-fix-problems-with-your-games/

Then uninstall and reinstall the App, and see whether it picks up your Sims 4 install.  Let me know either way.

Please also make sure your Steam and EA accounts are (still) linked correctly.  Open any EA website, for example ea.com, click the Sign In button, then the Steam icon, and enter your Steam credentials.  You should land in the EA account linked to that Steam account, which is also the EA account you need to use when signing into the App.  If you get an error when trying to sign into the EA website, let me know.

App Recovery from the Start Menu worked, thanks.

Though I took some extra steps, because I still had the "EA Background Service" running and I suspected it wasn't going to work if i just left that running, but also supected that it might be related to my most recent attempt at clearing the cache and therefore didn't want to just kill the process, just like that.

For anyone else who might have the same problem, those extra steps were: I rebooted my PC, exited the EA app, then killed the "EA Background Service" process via Task Manager. Only then did I start the App Recovery.

Rebooting my PC may  or may not have been unnecessary, but I think killing the process wasn't, and like I said I didn't want to kill it while it might still be occupied with another attempt at clearing the cache.