When I first enrolled in insurance I received written information from the insurance company regarding the use of that insurance. Included in that document was how to add dependants after a change in status to include marriage, birth, adoption. I did not expect to have a change in status but I filed that away in my brain just in case. When I planned to adopt many years later I went to Human Resources to ask how to add her. Knowing what is on your contract with the insurance company is each emplyoyees responsibility. Human Resouce Department is there to help you but they are not responsible for what you do or don't do. Your employer doesn't get involved in your health care. They may provide access to health care (not all do) but what you do to utilize that care is each emplyoyees responsibility.
You added ypour baby as a dependent for tax purposes. Doing so is unrelated to insurance. In fact the person who took the dependent information may be a different person than the one handling insurance information.
After MamaDuck's comment, she is right you are telling HR baby is a dependent eligible for insurance coverage. You then notify the insurance company of the addition. It would make sense for them to then tell you how to do that but the ultimate responsibility is yours.
My experience with medical offices is for them to confirm insurance coverage at first appointment. My latest granddaughter had her first appointment with pediatrician at 4 weeks. Did you then go to HR and ask for help or call the insurance company in a calm pleading manner? They may have bbeen able to help you then. Now it's too late.
I am sympathetic. Unfortunately this was an expensive lesson. Now you know to ask questions and not rely on others to take care changes in your life. Perhaps one of the most important lessons for me was the realization that I frequently thought I understood/knew things and therefore didn't ask questions. I learned to ask and ask and ask even when I felt stupid when I asked. I took care of the stupid feeling by saying something like "do I understand correctly that this takes care of the insurance. " no stupid feelings that way.