I just found out my daughter who was born with a broken clavicle now has Erb's Palsy?

When she was born her clavicle was broken i have talked about this about two months ago when she was born she was already in the doctors arms and i heard her clavicle snap and the dr made a big deal of it he was like did u hear that did u hear when i asked what it was he said it was normal it happens all the time then her umbilical cord was cut and he literally threw her on my stomach and said here have fun after she was born she cried for 3 hrs straight it was until i begged her pedi to do an x-ray cuz i just felt something was wrong, the found that it was broken, now two problems is if i wouldve brought her home not knowing that it was broken and then figured it out after and brought her to the dr they wouldve called DCF on me right or wrong?...after she came home she stopped moving her arm but she was barely moving it to begin with we brought her to childrens hospital in boston where they told me everything is fine i just recently found out the reason why she stopped moving her arm is because she has erbs palsy she has pt and neurology and everything now my question is, is the dr at fault?