"Unread news" feature went missing and I cannot get it back

"Unread news" is just a smartfeed. I guess you somehow deleted it. You can recreate it by clicking on File > New smartfeed and then making the smartfeed condition to look like "Status > is > Unread". By the way you can try out other smartfeeds, too. In fact this is the most advanced part of Feedreader :). Just search for phrase or combine feeds or show articles with some tags.