I'm sure that breast milk had a lot to do with it, but there are other factors, as well. How many little bugaboos is the child exposed to ... and the immune systems of the parents (genetics does matter) ... are the two that jump to mind right off the bat.
My son was not breastfed, mostly because it wasn't impressed upon me in the hospital to do so all those years ago, and that kid was so hungry that I don't think I could have filled him with milk in any event. He was eating cereal at 3 weeks because he wouldn't sleep for more than an hour or two because he thought he was hungry. Even though he never had breast milk, he was never sick as an infant and his first illness as a toddler was chicken pox. To this day, he very rarely gets ill.
One of my other DIL's, not the one I referenced above, could not pump enough breast milk to keep her youngest fed, and we had to add formula within days to get the baby satisfied. The lactation specialist assigned to this DIL/baby was equally as bad as the other one I wrote about. She had my DIL convinced that the baby would under-develop without "adequate" breast milk. Today, at 15 months, that child is far above and beyond others her age.
I'm glad that you threw the lactation nazi out of the room when she upset your wife. I know these women have a laudible purpose, but the way that some of them go about it is wrong and it does more harm than good. Evidently, bedside manner is not part of their training.
Not every woman can pump breastmilk successfully. I had no problem nursing my daughter but when it came to using a breastpump (and I had one of those expensive Medelas), I couldn't "let down" for the machine. So what I had to do was nurse my daughter on one breast and pump the other breast simultaneously and even then, I was only able to get 4oz or so. I could "let down" for my baby but not a machine and from I understand, that's not really uncommon. Fortunately, I didn't need to pump but I thought it would come in handy.
The other thing I wanted to mention is that newborns nurse a lot and they go through growth spurts where they act like they are hungry all the time. In one sense, they are but what nursing mothers need to understand is that by nursing more frequently, their bodies will respond by making MORE breastmilk to keep up with the baby's demands. It just seems really crazy for a few days while this process takes place. Unfortunately, if a mother supplements at this point, her body doesn't get the message that she needs to make more milk and then you get into a pattern where you really aren't making enough milk.
I read a book called, "So That's What They're For" and thank goodness she mentioned these growth spurts and the approximate times when they would occur because I would have probably thought the exact same thing and given up. As a matter of fact, my baby was born on a Tues and my milk didn't fully come in until the following Sunday. It can take up to a week but they get that colustrum in the mean time. My husband got a little mad when I chased him around the room with my new "squirt guns." lol