I did understand your point, and I could have been clearer by using the word "bill" rather than "cost". Obviously there was cost. In New Zealand cost is not something that a person suffering sudden trauma has to worry about. I am glad you agree this is a good model.
I have another story that is cancer related. Four years ago I had cancer surgery, in the private system. The public system judged that I could wait a month or two (our system isn't perfect), but I was fortunate to have health insurance and elected to be treated immediately. After a few days at home I awoke in the most excrutiating pain imaginable. I was just able to stagger to the phone to call for an ambulance. I would judge it was less than five minutes before I heard the wonderful siren of an approaching ambulance with well trained paramedics and their lovely lovely morphine. It turned out I had a relatively rare complication, a urinoma, my resectioned kidney was leaking urine into my abdomen. Ambulance care (google "Wellington Free Ambulance" it's a wonderful thing), ER assessment, CT scan, stent insertion to help the kidney heal, five days in hospital. Total cost: Who knows, probably astronomical. Total bill: zero dollars.
I love capitalism, I'm a true believer. But I believe even more in a compassionate society and it seems to me that means not using capitalism as a solution to all problems. It makes total sense to have a model that relieves victims of sudden medical trauma from financial stress and spreads the cost across society. The victims are not well placed to play the role of rational economic actors.