Needed a valid product code to registered to https://www.thesims3.com/registerag

I purchased the game through a subscription on Steam. In the game it is provided in product code which allows you to use the game.

I registered on the EA app and the game works without problems until you try to access the excange function to exchange content or download items.

When you press the dawnload button, the window asking you to enter the product key opens. But the one present in the game is not accepted (see attachment)

I contacted EA support who told me after an hour of waiting and chatting that the problem was with Steam.
So I contacted Steam who replied that the code is valid (and it is true because the game works) and to contact EA again.
Again an hour of chat to get a response to contact Steam. Do they play tennis?

The problem is actually simple to understand. The EA App allows you to launch the game without asking for any product code. But when you go to the site
https://www.thesims3.com/registeragame.html

The product key is requested which is not that of Steam but one that EA itself should provide.

So EA must somehow retrieve a code that is accepted by the site or modify the registration page. Otherwise the product is incomplete because the most important aspect of exchanging objects between users or downloading them from those made available by the game site is missing.
Need a technical intervention on the EA side is needed to solve the problem.
Regards Enrico