I have the same issue. I just contacted EA via chat, thanks to the poster who posted that link, and provided both of my game codes (I own two versions) and they added it to Origin so I can download, install and play it.
I recommend contacting EA via chat, they were quite helpful and quick. They were able to look up the game codes, see they were registered to me and add them to Origin without incident.
I’ve tried both and no luck. EA basically wouldn’t help me with origin. I have my original code and proof of purchase but since I can’t remember my email from 15 years ago they basically told me I need to purchase again. No wonder people are jumping ship for other games.