If you choose a pediatrician who has partners, the partners take turns being on call at the hospital. So, whichever one who is on call the day(s) you are there will be the one who sees your child.
If you do not inform the hospital who you have chosen for a pediatrician, then whoever the hospital has down as the "on-call" doctor will come see your child. That doctor could be anyone who practices from that hospital.
If that doctor doesn't take your insurance, you are SOL.
When my number 2 child was born, it was by non-emergency c-section. The hospital had a contract with an emergency care pediatrician, who was in the operating room. The hospital contract was such that this pediatrician practice was present for any anticipated "at-risk" deliveries. My insurance refused to pay until I filed an appeal saying I had no control over the issue.
With number 3, I delivered at Civista with Dr Miller's practice down as my pediatricians. I'd seen Dr. Miller the day before delivering because number 2 had a check-up. I told him I was having the baby the next day (another scheduled c-section) and he said straight out that he wasn't the doctor on duty at the hospital that day, so I wouldn't see him. But I did see Dr. Brubaker and Dr. Reed both during my stay.