If my child got hurt playing at someone's house I would take him to the hospital and I have medical insurance that would cover the costs.
That reminded me of the time my son was roller blading and did a flip and broke his arm. I took him to the doc and expected my insurance to pay the bill. They didn't. They grilled me about where and when the accident occurred, particularly if it had occurred on school grounds. Their intent was to find someone else responsible so they didn't have to pay. It was very irritating... my attitude was that it was a simply childhood accident and they should pay the darn bill. They eventually did but only after several rounds of correspondence. That was Aetna, btw.
And I guess my point is maybe some of the problem is with the insurance companies themselves.