I also think that such UI will be unnecessary, unless it displays something that you might want to see even if you are not touching it. Most commands can be entered by keyboard, including many commands for window management; some can use the keyboard and mouse together (e.g. to manually change the position and/or size of a window).

However, sometimes it may be helpful to display status indicators, e.g. if you have both tiled and floating windows, it can help to display an indicator for this. Also, an operating system design might involve capabilities, and the status of these capabilities (e.g. audio, network, automation, etc) might also be displayed as window indicators that the user can adjust (mute, redirect, etc).