Joseph:
First off, drunk driving is the preeminent problem. Don’t see how you can ignore it.
Second, it doesn’t have to be connected with driving. If the kid gets into a fistfight and hurts someone, or damages property, or anything else – you can be liable. Even worse, if c”v the kid walks into the street in a drunken haze and gets hit, you’re going to be liable. There are infinite liability possibilities, even without a car.