The Steam overlay isn't running during these times in the first place. No game is running, and no Steam window needs to be open. Steam is running in the background and its icon in the System Tray is visible.
This is entirely a conflict between a privilege-elevated UI (Groupy) and an unprivileged virtual device driver (Steam Input).
Testing this use case doesn't require having a Steam Controller, by the way. Any controller that Steam can recognize and configure can be used to test, from Xbox 360 and Xbox One controllers to the Pokken Tournament DX controller, provided you use a configuration that allows you to move the mouse and click.
