A time-intensive and physically demanding relationship would seem to be at a rather high risk for conflict. It's quite like being in an apprenticeship. I am pleased that my daughter had a gentle coach (at least as far as I know, as I preferred to stay attached loosely to avoid any overt meddling).
When performing the mentally and physically demanding task of skating, a skater can certainly be distracted and thrown off kilter by the daily variations in her moods and muscles. What if the coach asks your kid to do something and she's simply not feeling up to it that day? How much should the coach push, and how much should your skater push back? Who knows the best of what a skater is capable of, the skater or her coach? Who should decide the daily level of impetus -- and does the parent have a say in this?
Is there a good match between assertiveness in this relationship? Katarina (in her book on the subject) was actually pleased that her coach was stubbornly bossy and a hard driver.
Yet I saw one survey quoted that said "Only 7% of girls said coaches should be most concerned with winning" (although this survey was based upon a casual sport, something like softball I believe). I sense that skating kids have a different viewpoint than those participating in casual sports. Still though I would guess a good half of the skaters I'd see at freestyles don't really have intentions of ever competing beyond locally.
Here's a great page that describes many of the desirable characteristics of a coach. I'd grant a fair amount of leeway in each trait, and frankly the field you get to choose from is narrowed to those coaches taking new students at your rink. Bottom line: how you manage how pushy you allow (or choose) your coach to be depends a lot on your's and your skater's goals, considered together.