I agree with the OP on the large amount of blank space. This has been one of my biggest complaints with WP's UI. I understand that we can get "brain freeze" by having too much information available at once, but yet we soon become intimately familiar with our phones once they're set up. This makes brain freeze a non issue.
On mobile, space is at a premium. I've always felt that I do a lot more scrolling and swiping on WP than on iOS and Android, due in no small part to the large blank (read: wasted) space.
The problem here is not so much the hamburger menu itself as it is WP's implementation of them. Why do you have to tap directly on the menu to open it? You should be able to swipe in from anywhere on the same side as the hamburger menu to open it. It seems to me that it should be a relatively trivial matter to take care of. It's no wonder that WP users complain about hamburger menus. I hate them on WP, but I have no problem with them on iOS and Android.