Your argument comes down to "why don't we just let them die?"
You broke your arm? Sorry, no insurance. Figure out yourself how you're going to fix that. You might not be able to ever use it again, but that's your choice.
Cancer breaking down your stomach, leaving you in agonizing pain every hour of every day? Sorry, no insurance. Please be quiet, your neigbors didn't sign up to hear your screams.
Truck plowed into your car, leaving you maimed beyond recognition? Sorry, no insurance. We can't help you get home either, but there's a nice big hole outside the exit you could crawl into instead. Also here's the bill for cleaning the bloodstains you left on our floor.
We don't just leave people to die because it's both an ethical problem, and because it causes problems for everyone else.