Hello,
I've had the exact same issue with Mass Effect 3 for the past 2-3 weeks (I think my problems started on Feb 10 or 17, not sure).
The difference with the OP is that I purchased the game in December 2023, just a couple months ago. It was working all fine until I started getting the exact same activation problem as the OP. I get the same error windows and dialogues so I won't post screenshots. It's the same. In my case, I've never interacted with Origin and never had an Origin account. I've only played through the EA App.
I've tried submitting a case via the EA Help website but it seems to just go in a black hole. The system says that it registered my case but only gives a blank case number. Then, when I go look at my cases, it says I don't have any. I've already tried to submit twice with the same result.
Then, I tried to go on the Help Chat. That one just gives me a "Major Error. Please try again later" (I tried with Edge and Firefox, and please don't tell me to try with Chrome. EAGames should support all 3 browsers like every other major company on the Internet). So, it looks to me like EA games does not offer support to its customers at all. Just a bunch of webpages that look like a system exists but does not provide any service.
I must say this whole experience is extremely disappointing.
I hope something comes out of this thread and thank the OP for starting it.