Apple TV A1469 Wi-Fi Direct not available Everything I read (specifically: http://9to5mac.com/2014/09/23/peer-to-peer-airplay-apple-tv-ios-8/) states that the Apple TV A1469 running Apple TV Software version 7 and newer offers Wi-Fi Direct (aka peer-to-peer Wi-Fi) connectivity for Apple devices running iOS version 8 and newer so that AirPlay can be used without either device being connected to an access point.

This is almost exactly the setup that my family has - an Apple TV A1469 running the newest Apple TV Software version 7.0.3 (6917) and an assortment of iDevices (iPhone 6 Plus, iPhone 6, iPad Air, iPhone 5, etc) which all run iOS version 8.1 and newer.

However, none of these iDevices are able to utilise Wi-Fi Direct - AirPlay will only display if the Apple TV and iDevice are both connected to the same Wi-Fi network. See below for evidence of this.

Apple TV connected to the "Wiffy [2.4GHz]" Wi-Fi network:


iPhone 6 connected to the "Wiffy [2.4GHz]" Wi-Fi network:


AirPlay available on the iPhone 6 to the Apple TV, both connected to the "Wiffy [2.4GHz]" Wi-Fi network:


Apple TV disconnected from all Wi-Fi networks:


AirPlay not available on the iPhone 6:


Apple TV current software version:


Apple TV software up-to-date:


iPhone 6 current software version:
You have Bluetooth turned off on your iPhone according to one of your (many) screenshots. Bluetooth must be turned on for Peer-to-peer Airplay to function. See e.g. this Apple support article (scroll down to Peer-to-peer Airplay).