We decided to keep the hamburger menu on both platforms for launch. Our reasoning was that it's a common UI convention and our primary navigation options -- Home, Recommended, Hot News, Local News, and topics -- are visible in the extended app bar. An option to follow additional topics appears inline in the Home tab.
So the three functions that are only accessible through the hamburger menu are bookmarks, history, and settings, which seems like a reasonable compromise. You could use our app fully for a year, albeit with the default settings and no bookmarks/history, without ever seeing the hamburger menu.
Analytics shows that the hamburger menu is used frequently by our beta users, so I'm fairly confident that we made the right choice. On the other hand, the new YouTube Android app -- which had more in its hamburger menu than we do -- has moved in the opposite direction and eliminated it.
Inside the app, Home is a starting point that lets you see your news highlights at a glance. It does seem like the concept could be broad enough, as you say, to support history and bookmarks...
If Home is highlights of the day, then it seems natural to scroll down and see a heading for "Yesterday" with yesterday's highlights, rather than less relevant news from today (which could appear in the other views).