@Arishka145 I wouldn't have any idea what EA support has already suggested. But I do know that if you uninstalled the normal way, that wouldn't be thorough enough to take care of any Origin issues. Unless you specifically used Bluebellflora's guide to clean-uninstalling Origin, you didn't get the cleanest version of Origin possible. So please do that now if you haven't already.
Once you've reinstalled Origin, go into Documents > Electronic Arts and move the Sims 3 folder to the desktop. (This will prevent your existing saves and other user content from loading, but it can be restored later.) Download one Store item, something made by EA and NOT from the Exchange, and see whether it installs. If not, close the launcher and open it again. And also look for a second green plumbob in your Dock: this could be from the installer, hiding behind the launcher, and you can click it to bring it to the front.
I realize you've probably tried at least some of this too. But the point is to do all of it together and in one single test.