We do not have a free market. Most people get their health insurance from their employer - you cannot reasonably get insurance elsewhere (the same plan I have now for $100/month would cost over $1000/month, but I can't take that $1000/month to anyone else if I don't like my insurance so I accept it). This keeps me from taking a long leave of absence to live off my savings - until I quality for retirement (medicare).
In a true free market you would have more choices: most emergencies are close to home so you are likely to know which hospital is a good deal.
Of course there are still the emergency while on vacation, and the sudden collapse where a bystander brings you somewhere which are harder to handle. Though if there really was a free market there is incentive to handle those well - if only because the system is designed for the majority of cases where they have to handle it well or you would have gone elsewhere.
In a true free market you would have more choices: most emergencies are close to home so you are likely to know which hospital is a good deal.
Of course there are still the emergency while on vacation, and the sudden collapse where a bystander brings you somewhere which are harder to handle. Though if there really was a free market there is incentive to handle those well - if only because the system is designed for the majority of cases where they have to handle it well or you would have gone elsewhere.