You are a superstar @NastenaXD , thank you so much for helping me find the solution that actually worked! I've been in chat 3 times today with EA Help. I've done all of the "delete cache, sign out, sign back in, uninstall and reinstall EA app, try signing in from a different computer." Nothing was working and I was about to tear my hair out. The same thing had happened with Growing Together and what I tried last time didn't work this time.
I followed the instructions in the video to download and use Origin and was finally able to download Horse Ranch! As soon as I signed into Origin it verified my Sims 4 files, then started downloading Horse Ranch without any prompting from me. I think it's a HUGE mistake for them to not have the automatic download option in the EA app like it is in Origin. It's pretty obvious that the EA app is a total dumpster fire, buggy mess with new releases. If I have to do this every time there's a new pack release I'm going to be so frustrated.
But anyway, thank you for posting the solution that actually worked for me.