Ivy, the family cover you refer to wouldn't cover the OP's son in this situation. There is reference on the FAQs on that link to there being no cover for injury/damage caused by a child. Which makes sense as a child can't be deemed negligent. They also seem to place an age limit of 13 on children - again, this would fall in line with the age where children can often be found negligent/liable, depending on the circumstances etc.
It would be odd for anyone to offer cover for a situation where the subject being covered cannot be found liable. If you believe the cover you have, extends to situations like the OP's then maybe have a read through your policy because I do believe you are misinterpreting what the family cover you have, actually covers.
I dealt with claims like this about 10 years ago, so I am a bit rusty, but the general rule where a child has caused damage or injury is that anything under aged 13 is fraught with difficulty in proving negligence. Aged 10-13 in some instances, you might be in with a chance. Under 10, and you haven't got a hope in hell of proving negligence on either the child or the parent. And where you cannot prove negligence, there is no cover for a third party who has suffered injury/damage. Like I said earlier, if you are the sort of person who feels morally you have a responsibility for damage your child causes, then yes, pay for it. You won't get the cost covered by any insurance policy. If you are the sort of person who doesn't feel that you should pay for damage your child causes, the person who feels aggrieved by that would have no way to recoup their costs legally, so all the threats and posturing would lead them nowhere.
Hope that makes sense. As I said, it's been a while since I dealt with these types of claims.