I did not grow up riding and have been riding street for 3 years (I am 34). I have ridden mountain bikes since high school without injury (only rec riding).
I went through 3 cars in HS. Only 1 accident was multivehicle. Had I had a motorcycle then, I probably would have seriously injured myself. My parents didn't push me one way or the other.
When I have kids of my own, I want to take them riding with me, and have dirt bikes if they're interested.
To those who wonder at why street seems to be a bigger deal, here is my .02.
While riding off road and specifically recreational trail riding, almost all injuries will be a result of your own poor decisions or skill. On the street, no matter how good of a rider you are, there's always the chance of someone else's poor decision giving you a bad day. I feel everyone's been talking around this point without stating it directly. If the kids have been racing, then they also know to watch for others' mistakes and may have greater situational awareness.