The top four sources of calories in the average American child’s diet are grain-based desserts, pizza, soda and sports drinks, and bread. One-third eat fast food every single day. More than 90 percent don’t eat enough vegetables. And each year, our children are bombarded by around $2 billion in child-directed food and beverage advertising, much of which promotes the least healthy products.
If the kids bring their own lunches, those lunches will be crap. I can supply some anecdata on this. My mom taught high school and observed what the kids brought to lunch. I saw it when I did things like chaperone for field trips when my kids were in grade school. One kid's lunch was a sack of cookies. And if you want to be a snob for a while (disclaimer: I'm a bit of a snob), look at what people put in their shopping carts at the supermarket.
And it's quite noticeable that affluent people buy seemingly healthier food, and are much less obese, likewise for the lunches that their kids bring to school and the physical condition of those kids.
And it carries over into adulthood: A large fraction of my colleagues eat restaurant food for lunch, at least a couple times a week. The vast bulk of restaurant food is a pile of sugar, fat, and salt.
In my view, the US is not a fancy food country, and what works for us might not be fancy food, but teaching people how to prepare healthy food with modest ingredients. For instance I suspect that a sandwich and an apple would be a vast improvement over what most kids eat for lunch. That's all I ever got in the lunch that I brought from home as a kid. When I got old enough, it was my job to make my own sandwich.