According to official Micrsoft SDK, when you use back button to back out the app, it is telling the OS that user wants to close this app. The myth about double tap the back button is untrue.
So, there is no sense to argue whether or not a close app mechanism is needed because it is already offered for general apps. The only apps that are different are IE (which I typically close the tab before back out) and navigation apps. But if you are not actively navigating (with a route), these apps won't stay open either. And even if you do have an active route, once the navigation is done, the OS will shut it down after a certain amount of idle time.
If you use Windows button to quit an app, it will not be closed but rather being tombstoned which saves app state and suspended. No CPU resources are consumed while an app is tombstoned. So for those of you with OCD, learn to use the correct phone button