When you are raising kids, one of your larger concerns is how to choose the quality of the adults that they get exposed to. For most parents a lot of this concern gets played out in the evaluations they go through when choosing a school for their kids. It also influences the types of friends you encourage them to hang out with: by association the attitudes of their parents filter through their kids. When your kid attends school, they get a new teacher every year, and once they reach middle school they get several teachers every day, and then a whole new slew each semester. In total your kids tend to spend more time with adults who are teachers than with their own parents.
When your child skates competitively though the only adults she spends a lot of time with are her coach and perhaps her ballet teacher. These usually remain the same people year after year. Choosing the coach that matches the personality of your kid and likewise meets your desires for the proper role model therefore gets to be a bigger deal.
Oddly, skating or coaching ability tends to fall out of the equation; unless your kid has demonstrated national abilities at an early age, any coach you choose from the rink who has skated competitively herself and who is a member of PSA will have adequate skating skills to teach for local competitions. Therefore spend a bit of time when your kid is little to soak in how the coaches interact with their students, from a personality perspective. You are choosing those ideals in an adult mentor that will serve your child well beyond her skating career.