I would have gone to the pediatrician and not the ER. So I agree with where you took her. (I took my daughter to the Pediatrician with a broken leg...I was hoping for a sprain, but it was broken). There was no swelling, bruising or redness on my daughters leg...
On your situation...I got curious and Googled nursemaid's elbow and found the following...on WebMD...
Causes of Nursemaid's Elbow
Nursemaid's elbow can happen if you tug or pull on a child's lower arm or hand, especially if the arm is twisted. It doesn't take much force for the injury to happen. The most common cause of nursemaid's elbow is a pulling-type injury.
Nursemaid's elbow may happen if you:
Catch a child by the hand to stop a fall.
Lift a child up by the hands or wrists.
Pull a child's arm through a jacket sleeve.
Swing a child by the arms or hands.
Yank on a child's arm to make him or her walk faster.
Sometimes nursemaid's elbow may happen if:
An infant rolls over onto the arm.
A child uses the hands to brace himself or herself during a fall.
Symptoms of Nursemaid's Elbow
Nursemaid's elbow can be quite painful. However, the arm or elbow doesn't look hurt, the way a broken bone or sprain might. There is no swelling or bruising.
Symptoms of nursemaid's elbow include:
Arm hangs straight down on side of body
Child is not using the arm
Pain when moving the arm
Pain in the elbow, wrist, or shoulder
Treatment of Nursemaid's Elbow
Treatment depends on your child's age and overall health. The doctor will examine the child and make sure the bone is not broken. X-rays are not necessary to diagnose this.
Over-the-counter pain medicine, such as acetaminophen (Tylenol) or ibuprofen (Advil, Motrin), may be given. Make sure you ask your health care provider for the correct dose for your child. Never give aspirin to a child under age 12.
So it sounds like from WebMD the doctor was acting in a proper manner for what she thought it was and that she didn't need to x-ray it first. However, it seems if you told her the injury was from a fall and not having her arm jerked then I would like to think she might have done an x-ray first just to be on the safe side...
I would give the doctor the benefit of the doubt...I know it was very very painful...but the exact same thing could have just as easily happened in the er as it did in the doctor's office. If an experienced pediatrician made the mistake so could an intern or first year resident in the er.