Check if the YT icon is not simply hidden in Parental Controls.
YT App for Apple TV is integral part of Apple TV's iOS bundle, so there's no way to install it separately afterwards.
Where did you source such a device? You want to say, that after unboxing a brand new aTV, it didn't have any icons on it's opening screen?
Try firmware restore via iTunes, using USB-to-MiniUSB cable.
PS Be sure the aTV is not powered on during this excercise!
PPS I've never come across installing profiles on aTV, but according to Apple, that should also be possible. Not sure if you can config anything other than network via profiles, though.