The stuff that I bought at the Sims 3 Exchange store won't download.

kepkesr  Please try clearing Edge's Internet Explorer-mode data, as described towards the end of this article.  Even if you never use Edge, the launcher does for some purposes.

https://support.microsoft.com/en-us/microsoft-edge/view-and-delete-browser-history-in-microsoft-edge-00cf7943-a9e1-975a-a33d-ac10ce454ca4

Delete the file labeled [your email].bin in Documents > Electronic Arts > The Sims 3 > Downloads as well.

If that doesn't help, please create a new admin Windows account and try installing the content there.  Make it local, as in, don't link it with your Microsoft account or any email address.  Your saves and other user content won't be available, but that can be addressed later; for now, just see whether you can install the Store content without getting the login request.