> competition always leads to lower prices

I don't see how this could be true for emergency visits. Would an ambulance drive you to the cheapest hospital within some fixed radius?

Hospitals typically lose money on emergency visits and make it back on scheduled inpatient care and outpatient services. This would accelerate a poor performing hospital's demise, because ambulances will go to the closest one but patients who have options will look elsewhere.

If you now have two ERs within driving range, you have the choice to go to the cheaper one if you are conscious and in a stable enough condition to reflect. This is the sort of thing people already think about in the US.