After getting anxious, and not finding any other helpful information on the forums or web, I decided to just bite the bullet and do a complete reinstall after downloading WordPress again (after all, there was really no content on my page yet).
For whatever reason, it worked. And I now have the “Add Media” box when I click the button.
Not sure what caused this. Both times all the files were uploaded successfully to my FTP.
Oh well… it’s working now.