I find it strange (considering what a huge issue obesity and the health effects are) that the doctors think the baby will "out grow" being overweight (most studies I've seen show that is not true) and they aren't concerned with this or viewing it as a problem.
I don't think it's strange at all. He's four months old. The studies that you're probably thinking of were regarding toddlers, not infants. By the time a child is 2-3 years old, they've already developed eating and other habits that could contribute to a weight problem. So, if at that age they've developed a problem, of course they're less likely to outgrow it. However, at this boy's age (and per the original article), all he eats is breast milk. They're not giving the kid cookies and Big Macs. And there are *plenty* of large, breast-fed infants that grow into healthy, active, 'normal'-weight toddlers, children and adults. My best friend's son was one of them-- he was a large baby at birth, quite chubby as an infant, and now at 23 months, he's still at the top of the charts for height, but is so skinny she has trouble finding pants that stay on him.
The thing that strikes me about this issue is that people refer to the health problems of obesity, and yet the discussion seems to focus on the boy's appearance. The concern, particularly with regards to children, shouldn't be weight, per se, but nutrition. If you have an 'obese' toddler, the concern should be on whether they're eating a nutritionally balanced diet, and on whether or not they're developing problematic eating patterns. (Do they refuse to eat anything that isn't frosted? Are they constantly given snacks (healthy or otherwise) in order to placate/bribe them?) These considerations simply don't apply to infants. Therefore, it makes perfect sense for the parents and pediatrician to be unconcerned with the baby's weight.