This is classic marketing and fundraising advice, which is what I do in my other life. You need to engage your potential customers, or they'll go with the person who does engage them.
My story is summed up in my tag line: "anyone can learn to skate." I've got a good story as an adult skater who loved the sport so much that I changed my life in order to pursue the passion. Looking around my rink I see some other good stories-- the young coaches who grew up at the rink. The grade school teacher who coaches during her off hours. The Russian stars (at least according to them, but whatever, it's a good story), "my dad's a former champion," and of course, the former champions.
Then you need to ask yourself: why does this story resonate with me? What is it about this story that makes this person a good coach? Does the answer change if you think about different skaters-- for instance, the former champion has a great story, but is that the right story for the recreational or timid skater? "My dad's a champ" is a great story, but do that coach's skaters actually get to skate with dad? In other words, is the coach adapting aspects of the story to reach your individual skater's needs?
What's your coach's story? What coach do you know who has a good story of their own?