I think it's because families with 1 to 2 children are often trying to make life easier for their kids by giving their kids what they never had. It's like perpetual spoiling for most of them.
Parents who have a large family, on the other hand, cannot afford the time or the money to spoil their children.
Of course, that sounds like a smaller family would be preferable over a larger family, but that's not the case most of the time. The people who grew up during the Great Depression and later went to fight in WW2 (or supported those who went to war) were the last generation to really try and make a living in America. But they raised their children so that they would have the things they wanted (and not needed), and parenting completely changed in America.
What most of us don't want to believe is this: Children need to struggle. They do. I don't mean they need to be neglected or malnourished, but handing children everything they ever want is not preparing children for reality. Children who don't get the latest and greatest, who don't have the assurance that their parents will pay for college, really learn to rely on themselves instead of expecting a perpetual adult to be there for them, whether it's actual parents, a boss, a sergeant in the military, or the government.
I was one of two kids, by the way, but my father always made sure we worked around the house and in the yard. I hated it at the time, being the kid who didn't get the latest designer jeans and the only one of my friends who actually had chores, but I grew up with an amazing work ethic and a lot of small skills that have really helped me in life. The first job I ever took was dishwasher, and while a lot of my friends laughed at me for taking such a "pathetic" job, I was the one who was able to afford having a car of my own that my parents couldn't take away when I was grounded.
So I guess that's the situation. It really doesn't have much to do with the number of the kids but the attitude of the parents, and a parent with multiple children and limited resources is going to have more of that "old school" parenting mentality because it will be the only one that works for their situation. Parents with few children could replicate this by actually expecting their children to contribute (appropriately) to the household and not making everything soft and cushy.