Captains are responsible for the overall safety of all occupants of an aircraft, but they are not responsible for the care of unaccompanied minors. That is a cabin crew function, and may mean that cabin crew require a higher level of screening (though I've heard nothing from our cabin crew to suggest that is the case).
As a pilot, you are behind a locked door for the flight and have nothing directly to do with any passenger, let alone unaccompanied minors. I would be extremely surprised if any airline required more than the basic disclosure. Virgin Atlantic certainly does not.
Scroggs