New APIs in Android 4.4 KitKat, as I understand it, allow you to have true full-screen, where though the status bar and toolbar are still visible, your app can draw content underneath them. I suggest that we at least experiment with using these APIs, in combination with viewport margins, to draw page content in this area. I imagine a proof-of-concept wouldn't be *too* difficult to get going.
Rather than draw content under the nav and menu bars, I would recommend making them disappear in the same manner as the location bar. A perfect example of this is Pocket app's webview. Scroll down and you see nothing but webpage. Scroll up and location, nav, and menu bars reappear. It's beautiful.
I actually just want fullscreen mode (videos, reader mode) to use this at first. Doing what comment 5 describes sounds awesome too, but that would be 2.0, IMHO.