By default, category and tag archives only list Posts in that category. So, despite the UI appearing on the backend, the posts don’t show up in the archives. Luckily this isn’t too hard to solve.
Look at this support thread and this answer in particular. It has resolved this issue for me in the past. Make sure not to use the first few code snippets in the thread as they interfere with menus.
You may also want to look into Justin Tadlock’s post: “Showing custom post types on your home/blog page”.