No one satisfies customers all by themselves. Even the brightest and most dedicated employee needs the cooperation of other team members and departments to accomplish this.
It's good to be individually brilliant and to have strong core competencies. But unless you're able to work in a team and harness each other's core competencies, you'll typically perform below par; for there will always be situations in which you will do poorly and someone else will do well.
Teamwork is more than just showing up to work at the same time as others, then working in the same vicinity all day and hoping to somehow achieve some vague perception of relevancy. Teamwork is mainly about situational leadership, letting the person with the relevant core competency for any given situation take leadership. Egos must invariably be set aside from time to time (keeping them at home is even better).
This can be the most challenging facet of teamwork: allowing someone else to "get the credit." And yet, this is how teams accomplish more than any one person can do on their own. Every person recognizing that they have a piece of the overall success of the team creates a vibrancy and contagious enthusiasm that helps keep the team moving forward. After all, it was Mark Twain who said, "It is amazing what can be done if no one minds who gets the credit."
True teams understand that customer satisfaction stems from a consistency of customer experiences over time in interactions with many different people, not just from one encounter where things happened to go well. Keeping this in mind will help to provide a sense of urgent consistency that every interaction with customers is equally important, and potentially impactful in positive ways.