The SHIELD is only be able to listen for devices that can pair to it. It can't scan for Bluetooth devices and pair with them, such as headphones. Why did they miss out a very basic Bluetooth feature? Is there an app to bypass their silly accessory page and just scan?
Even Chinese boxes can do this properly. Sigh.
Huh? Shield TV can definitely connect to bluetooth devices; it had this even before the Nexus Player (it was backported from Android M). Plenty of people around here use bluetooth mice, keyboards, remotes and headphones.