Won't let me play, telling me I NEED to use steam in order to play

Enigma_nine​  Try claiming the free holiday pack in the EA App.  That should add Sims 4 to your EA account.

If that doesn't work, you could temporarily unlink your Steam and EA accounts, or temporarily remove Sims 4 from your Steam account.  Either way, clear the EA App's cache afterwards:

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

Then you should be able to claim the base game since the App won't think you own it anymore.