@mottle To answer your question, no, there's no way to bypass the EA App, and no legitimate source for downloading the pack that doesn't involve the App or Steam or Epic. (As a side note, the pack doesn't have its own .exe; the only excecutable is the one for the base game, TS4_x64.exe.) The App should show that you own Horse Ranch though.
Before doing anything else, please clear the EA App's cache and restart your computer.
https://help.ea.com/en-us/help/faq/clear-cache-to-fix-problems-with-your-games/
Then check Manage add-ons again. If you still don't see Horse Ranch on the list, and please check the entire list in case the list isn't completely in alphabetical order, go to ea.com, sign in, open Account Settings, then Order History, and check the status of the order. If it says Processing, that's the issue, and you'll need to cancel the order and try again.