I can see the problems of allowing poorly equipped and untrained people to participate. But I can see even bigger problems (for the SAR team) if they tried to physically stop the family members of a lost child from searching. If the child were later found dead, say from exposure from being out too long, I can easily see that SAR team being sued out of existence. I would probably sue them myself if it were my kid.

A tough call for SAR. But I think it would be best to try to channel the family's participation into the most fruitful and safest activities, but barring that guidance being accepted, I'd think it very unwise to restrain them from participating. Extreme cases are different. I can't imagine a fireman allowing a family member to run into a burning house to save someone. They're probably reasonably safe from liability in preventing that kind of "family participation".