I do not have enough knowledge to comment on your child's case. But in general, a general pediatrician more or less serves as a gate-keeper -- taking care of the small stuffs and watching out for the dangerous stuffs. If they do a good job at that, it's fine.
One of my friends had a very complicated delivery and her baby came out with a bad skull fracture. The pediatrician looked at the MRI and told my friend the baby had a massive stroke and proabably would never be able to walk or talk. So she was sent to a specialist. It turned out there was no stroke. The specialist commented that general PDs would not have had much experience looking at babys' brain MRIs. That's OK as long as they know when to send the patient to a specialist.