In the course of parenting a skating kid, our family has seen some family interactions in passing. Everything from absentee parenting to career-ending smothering with little outside intervention. A couple of times I've seen a parent (in different disciplines) run through the coaches, repeatedly burning bridges for their athlete. Or just coach-jumping to an astounding degree.
Part of the problem is, if your child does not train at an established center and isn't a natural phenom but building their skills over time, then you can't help but have to be involved, at the start. The family has to find the on-ice and off-ice coaches and training facilities to develop the skater.
Plus, if you are a skater and practicing to pursue pairs or ice dance, unless you have a partner, you lose out. In the US, there are partnerships at the Juvenile to Novice level that can access development camps that more talented but unpaired skaters cannot. So the family has to cobble together a training regime. Even if you have competent to excellent coaching, there is a gap. There are some highly-specific things my skater needed/needs that coaches still have not quite been able to help with. It's frustrating to be in such a position.
Also, you could be at a large, popular and successful facility, but if your coaches aren't the top ones at the facility, there might be things that your skater can't access - for example, special jumping rooms. So there are skaters with good coaches at good rinks that can't quite have the same things as the best skaters have. Which can become a self-fulfilling prophecy.
Ideally, as the skater moves into the Novice/Junior level, the parent can be confident that their skater is older, well into their skating career, and able to be more independent. (I'm thinking more about personal safety issues in general.) But our family has also seen some coaching that had unhealthy boundaries for older skaters where the situation didn't feel appropriate - skater is technically an adult, parents and federation are abroad, and the skater is isolated in the situation. Could have benefitted from parents being around more. This was with two senior-level skaters. Sad all-around.