There is no mention of attaching weights to the significance of each factor of success.
I don't think you'll find many objective and reasonable people who disagree with the main point. In my opinion, your interpretation takes things to an obvious extreme.
Going back to a football analogy (noted: you've already shot down the premise), Aaron Rodgers attains a tremendous amount of individual success by setting passing records and winning Superbowls. Nobody including Obama denies this individual success which came from hard work, talent, and drive. However, this success is not attainable without an NFL football field, NFL rules, officials to enforce these rules, fellow players, current and former coaches, and consumers to watch and fund the sport.
This is the real world. We rely on each other to provide an environment where individual success is a possibility. In no way does this statement diminish the value of each individual's actions toward success. If we lived in an "everyone for themselves free for all," modern societies would break down, and the world would be a complete mess.