The thing was, from an outsider's point of view, not that the coach wasn't part of the equation but that the coach became the major element of the equation: it sometimes appears (especially during the Medvedeva kerfuffle and then Trusova's less fraught exit stage left) that a number of fans have elevated Tutberidze as the star and idol above her pupils and for those fans, the skaters' duty is to burnish her place in their pantheon. Which annoys the c*** out of people who believe coaches are there to do the burnishing thank you very much and it's mainly, even only, the one out there on the ice under all the hungry eyes who matters.
Me, I think neither is extreme is right, and the middle path is probably best. Tutberidze deserves lot of admiration and kudos as does three-OGM Brian, as do other coaches. As will Plushy if he does good by Trusova. But when the skaters begin to blur in the public gaze and just become #TeamWhatever, they are more than entitled to not care for it very much.
After all, in the history books, it will be Zagitova and Hanyu and Virtue/Moir and Dick Button and Plushenko-the-skater who will be inscribed most brilliantly, not Tutberidze and Orser and Marie-France Dubreuil and Gustave Lussi and Plushenko-the-coach, and that is IMHO as it should be.
It's not only Eteri's status and it is not new phenomenon. Names like Mishin, Tarasova, Moskvina, Morozov are known sometimes more than their pupils. With Eteri it probably just got more international recognition. Once again I will use the czech TV commentators an example. They often do not speak much about particular skaters but about Eteri's team (mostly not in a very positive way, like "another member of the Eteri's hatchery" etc. ). And when you mentioned Moir, I can't forget how precisely him spoke about Alina as "that russian girl who is only jumping".
As for me particularly, I admit that I am strongly influenced by my way from a person with zero knowledge about the sport to a devoted fan. At the beginning I saw some performances by chance and later, when the interest started to grow, I've noticed that all the skaters, that catched my attention, were coming from one team. And, as a paradox, I can say that when I started to learn more about the sport, skaters, rules and coaches, I believed the negative comments about Eteri and had a negative opinions about her and only as I've learned more to the moment when I started to make my own conclusions, I've changed my mind Whether it's the recognizable style or the technical perfection it's not important now, but I liked what they did more that what the others did and made me a fan of them all
Still, even if I am a fan of the team (no reason to deny that ), that doesn't mean I don't recognize the skaters as individuals and don't have my preferences about them. Though it is sometimes hard to choose. That also doesn't mean I don't root for skaters who are not part of the team anymore. I still support Nastya Tarakanova, I wished all the best to Panenkova etc. and I don't think that people who went to competitions with banners using the abbreviation 3A or something similar that they were or wanted to be insensitive. As I have said, it is sometimes hard to choose the favorite among them. I can't speak for others but for me, when I talked about them all together, it was more because I didn't want to leave some of them less noticed than the others