Just a minor correction:
The payment that insurance companies make toward emergency treatment of road traffic accidents is totally irrelevant to this topic, since it is always paid in the first instance, not by the patient but by the person causing the injury, whether or no that person is culpable.

Of course, there will be a large number of occasions where the patient & the person causing the injury are the same.

Break your leg on holiday abroad and you need to rely on holiday insurance to cover your hospital costs. Do so while drunk and they may refuse to pay up.
Why shouldn't we have similar access to services over here?