You have an over-simplistic view of the original research. It wasn't only about the "willpower", but also coping mechanisms than we employ to control our urges. And it proven that some children are way better in this than others. For instance some kids would turn their back to the cookie, or would come up with some other distraction, so that they don't think about eating it. Having this type of resourcefulness to quickly adopt to the situation is probably a personal trait that comes very handy in one's life.
You’re arguing definitions. Resourcefullness is encompassed by willpower in the context of this article.
Yet the point I am trying to make is that poor kids would have a tendency to make their decisions based on their context which isn’t, sadly - optimal in most cases.
The connection between “this kid doesn’t have to think about rudimentary issues because he’s better off” becomes closer to “rich people can risk more because their risk is hedged by their well-being”.