I hope it's okay to update here -
I found a fix to launch, using the Sims 3 Alder Lake processor patch from LazyDuchess from this discussion worked for whatever reason. I don't know why it was suddenly needed, as I am using a newer compressor for a few years now, and the game only started acting up this month. But whatever the patch did seemed to fix the launching issues, at least for now.
I still had not had the EA support return or explain what is it they saw was wrong with my registration of the game, and the EA app's library still links me to 'Purchase the game' as the screenshot I attached to my earlier post showed. Hopefully it won't cause any problems in the future.