Yes but emergencies are the exception, not the norm for most hospitals. Most hospitals make 40-60% of their revenue on elective surgeries, ie things that can wait 2-6 months. So patients have time to shop around.
Emergencies are the norm, not the exception, for most people. Elective surgeries are mostly the well-insured and the wealthy. Poor working people go to the hospital when it hurts so bad that they're missing work, and when they miss work they don't get paid.
Source: currently work at a big hospital in IT.