If I am understanding the issue correctly, (Can't access downloadable content in-game.) You may need to turn on Origin In-Game.
In the Origin client, click the word Origin > Application Settings > Origin In-Game - Click the off/on button to turn it on.
There is a known bug with Origin In-Game not working properly. I can't tell if the issue has been resolved yet or not. I also am not finding anything much on that part of the problem.
I am assuming you can view the in-game gallery just fine?
If you want to see what packs you own you can go to the Sims website:
https://www.thesims.com/ and click on Store > Visit the Store. After signing in, it will show you what packs you own and/or have to purchase.
You can also see what packs you own in the Origin client by clicking on My Games Library > Sims 4. Under the Play Button look for the words Extra Content. Click on that to see what DLC you own as well as the freebies.