Then where do the stories about a family getting hit with a $100,000 hospitable bill come from?
Oh, I've got stories too, from my own family. When something like this happens, you talk with the doctor or facility, and you work it out. You'd be surprised what they can do from payment plans to write-offs.
Back when my grandmother was diagnosed with breast cancer (that eventually spread to her lymph nodes), $100s of thousands were racked up trying to save her life. She had no insurance other than Medicare, and Medicare covered less than 50% of her expenses. After little discussion, every single doctor and medical facility took write-offs for the unpaid charges. She (her estate) had to pay NOTHING.
In your scenario, little Timmy's parents will get a bill. They may even be requested to sign paperwork affirming their responsibility for the charges. In the end, they are responsible for caring for little Timmy. If they didn't make the appropriate decisions beforehand to ensure that he had adequate health coverage, they need to be responsible now. If they can pay the bill, they should pay it. If they can't, they should negotiate it.