Hmm.. I tried the Exchange the other day without issue, but haven't checked the Sims 3 Store side of things since a little earlier this year. Let me give it a try and I'll post back with results.
Edited to Add:
The Sims 3 Store (
https://www.thesims3.com/home.html) website seems to be working for me. When I go to download anything from the store or exchange for that matter, I create a new Sims 3 folder for easier troubleshooting/testing.
Sims 3 -
How To: Create a New Sims 3 Folder - With Photos
I don't add anything back to the new folder. Just use it strictly to help download whatever I'm after from the store or exchange.
I open the launcher and then log into the store or exchange. I find it easier and faster to open the launcher first so the launcher is ready to download whatever it is you want to download.
If you have links to what you want to download ready, go ahead and use them to get the items. Click the Downloads button on the Store set or item page and for Exchange items, click the add to game button.
Here's a few things I just downloaded for my test: -- These files downloaded fine.

– May need to make sure to allow the store to open Sims 3 links. It will open the launcher and start downloading the .Sims3Pack file(s).
Frontier Finds Kitchen and Dining
https://store.thesims3.com/setsProductDetails.html?productId=OFB-SIM3:35696Contemporary Comfort Bedroom
https://store.thesims3.com/setsProductDetails.html?productId=OFB-SIM3:37312Nautical Living
https://store.thesims3.com/setsProductDetails.html?productId=OFB-SIM3:65741Japanese Inspired Living Compilation
https://store.thesims3.com/setsProductDetails.html?productId=OFB-SIM3:44628 -----
If for some reason you can't get store or exchange items to download, or get the Don't Own Item "bug" close the launcher and try either a different browser and/or clear your browser cache.
How to Clear Your Browser's Cache
https://www.wikihow.com/Clear-Your-Browser%27s-CacheDon't forget to clear Edge and IE's Browser Data:
The Sims 3 launcher uses parts of Internet Explorer to work, so clearing Edge and IE's browser data can help with launcher issues such as website issues, downloading and installing store content and issues with exchange downloads.
View and delete browser history in Microsoft Edge:
https://support.microsoft.com/en-us/microsoft-edge/view-and-delete-browser-history-in-microsoft-edge-00cf7943-a9e1-975a-a33d-ac10ce454ca4If your password contains any special characters, it's a good idea to change your password. The launcher doesn't do well with some of those, for example & or !. This is the case even when the password works fine on other EA sites.
Additionally, go into Documents > Electronic Arts > The Sims 3 > Downloads and delete the file labeled [your email].bin. if you have it.