Ezra Klein points to an obvious reason they are always open:
One of the big problems here is time: Primary care doctors don't have much of it, and what they have is generally between 9 a.m. and 5 p.m., Monday to Friday. Patients also don't have much time, but what little time they do have is between 5 p.m. and 9 a.m., Monday to Friday, and then the weekends. Emergency rooms have long waits, but at least you can schedule when you're waiting for a time when you're not supposed to be working.
It's things like this that add up to the kind of inefficiency an allegedly more efficient market-based healthcare system is supposed to provide. Why not an emergency assistant at your doctor's for 24-hour help, or even just a phone call?