I think the distinction we're trying to make is between compensatory damages (which are intended to compensate the plaintiff for the actual injury), and punitive damages, which are intended not to compensate the plaintiff, but to punish the defendant. The purpose of punishing the defendant is served whether the punitive damages go to the plaintiff or someone else.

Compensatory damages normally include such things as economic loss, pain and suffering, loss of companionship, loss of enjoyment, etc. So if you proved that someone negligently caused a severe, disabling injury to you or your child, odds are good that you'd get enough money to make the best of things even without punitive damages (assuming that the defendant is adequately insured and that the state hasn't placed limits on pain and suffering awards). Some states don't even allow punitive damage awards.

So, assuming that the plaintiff and his lawyer have been adequately compensated for the injury via a compensatory damge award, a reasonable argument can be made that a punitive damages award should not result in a windfall to the plaintiff and his lawyer, but should be applied to court costs and other higher purposes.