Maybe I do not understand what you are asking, so forgive me if I miss the mark.
Obviously a competent coach is going to be better than an incompetent one. A coach who cannot communicate appropriately or effectively with the age level they are coaching is certainly a problem. An abusive or cruel coach should not be coaching. I would also agree that the overall philosophy of a program matters.
But here is what my experience is- my 2 sons have been on the same team for 6 years. We have seen multiple coaching changes and one HC change. Some coaches have been part time, some full time. Most pretty young, but some experienced and some inexperienced.
Both of my sons both did well with every coach. They liked every coach, and they improved under every coach. Several times over the years we have heard of other gymnasts (or their parents) being unhappy with this coach or that coach: This coach is too mean. This coach yelled at me. This coach is not keeping discipline. This coach is not tough enough, and also, "my kid is not doing as well as he should under this coach." Kids complained to parents, and parents complained to the gym, and sometimes coaches left due to this, and my kids (and I) never knew why. Again, they liked and did as well as they are capable of with every coach they ever had, and I believe they learned important things about gymnastics-and life- from every coach they had. Most importantly I always felt they were safe and they always have had fun. 6 years and neither ever wanted to miss a single practice.
I guess I am saying that I think that of course the coach matters but the athlete's attitude and ability to listen, behave appropriately at practice, and to be coached matters too.