"The rest of you can go on ahead," they said, as if we mighty fly on despite the loss of a teenager.
Why would that be a problem? You're talking about someone who is close to the 18 end of teenagerhood, but even so, so what? I was hitch-hiking form one end of the country to the other when I was 14 (which was about 11 years before I first flew in a fixed-wing aircraft). Make sure the kid has a charged mobile phone, get on the plane, sort him out with a hotel room by phone, and then get on with sorting out how to move him on to catch up with the rest of the family.
Beside, whose fault is it that the person hadn't got appropriate ID? Yours, or the young adult? They're often whinging about not being treated like adults, so ... here's your power, take the consequences if you get it wrong.