Unfortunately yes this does happen a lot (it happened to my oldest nephew) and it's often the doctors fault because they are not properly trained in how to resolve a shoulder distocia (and has nothing to do with the size of a woman's hips but in the position most women are forced to labor and push in). Proper technique would be to flip a mother onto her hands and knees or into a squatting position to help open up the pelvic and allow the baby to pass. Most mom's are pushing on their back which actually reduces the pelvic opening by 20%. Once a baby is stuck the doctor gets scared and starts pulling on the neck and often break the clavicle to get the baby out fast. This can often result in a birth injury where the nerves are damaged in the neck and numbness can happen in the arm. A lot of the time the baby is fine though and it heals very, very quickly (6 weeks or less). Unfortently due to such a large number of mom's using epidural during delivery a position change is often out of the question (though even turning them onto their side would be better then on their back) so a better technique (and one that is taught to and practiced by midwives in a true stuck situation where a position change doesn't resolve the SD) is to hook two fingers under the shoulders and pull. This prevents the neck injury and still get the baby out. My OB used this technique when I found myself in the hospital for my last birth stuck pushing on my back (the doctor put me there to help hold a cervical lip back so I could start pushing and I didn't have the the mental capability to think that I needed to change positions in the middle of pushing because I was so caught up in the moment) and my son got stuck.The doctor only used one finger though so my son's arm broke (two fingers distributes the pressure over a larger area and greatly reduces the chance of this happening). It was pinned just as you described for your daughter and he was fighting the restraint my 5 weeks. You would never know that his arm was broken looking at him now at 14 months.
Updated
Unfortunately yes this does happen a lot (it happened to my oldest nephew) and it's often the doctors fault because they are not properly trained in how to resolve a shoulder distocia. Proper technique would be to flip a mother onto her hands and knees or into a squatting position to help open up the pelvic and allow the baby to pass. Most mom's are pushing on their back which actually reduces the pelvic opening by 20%. Once a baby is stuck the doctor gets scared and starts pulling on the neck and often break the clavicle to get the baby out fast. This can often result in a birth injury where the nerves are damaged in the neck and numbness can happen in the arm. A lot of the time the baby is fine though and it heals very, very quickly (6 weeks or less). Unfortently due to such a large number of mom's using epidural during delivery a position change is often out of the question (though even turning them onto their side would be better then on their back) so a better technique (and one that is taught to and practiced by midwives in a true stuck situation where a position change doesn't resolve the SD) is to hook two fingers under the shoulders and pull. This prevents the neck injury and still get the baby out. My OB used this technique when I found myself in the hospital for my last birth stuck pushing on my back (the doctor put me there to help hold a cervical lip back so I could start pushing and I didn't have the the mental capability to think that I needed to change positions in the middle of pushing because I was so caught up in the moment) and my son got stuck.The doctor only used one finger though so my son's arm broke (two fingers distributes the pressure over a larger area and greatly reduces the chance of this happening). It was pinned just as you described for your daughter and he was fighting the restraint my 5 weeks. You would never know that his arm was broken looking at him now at 14 months.
I should add that the reason that a child has no issue with a bone break, as long as their is no nerve damage, is because their bones grow so much that by the time they are adults the bone that was broken will have become the marrow inside the adult bone.