D.W.
No, this is not normal because of the number of children. It's just the bedside manner of the physician.
I spent the last 10 years in and out of hundreds of physician offices professionally - all have different personalities and different levels of engagement in their patients. Their biographies may be impressive, but education and years of experience don't translate into bedside manner.
That being said, I was diagnosed with cancer when our second was 10 weeks old. My oncologist is the best lymphoma expert in Indiana. I did not like his bedside manner, but I'll take his brilliance any day (his nurse is an angel).
If you don't like her bedside manner, and she's not providing the care you expect, you have every right to find another pediatrician or a family practitioner (though I personally recommend a pediatrician at that age).
If you're in the situation again where you are not covered by insurance, please know that just about every community has clinics that service people without insurance with fees on a sliding scale based upon ability to pay.
Our kids are 22 months apart - it's really critical to ask questions when that second child is in for their well-baby visits. In our case, the American Academy of Pediatrics had changed their policies on several things between children. I'm glad I made a comment about introducing certain foods (with our older child we were told NO peanut butter before 15 months). The AAP had changed their policies in that period of time.