I had a lot of different coaches, who advocated very different techniques. It can be confusing.
I employ a few different coaches between myself, my spouse, and five children. Having one coach for everybody just isn't feasible due to scheduling constraints. There's definitely a divide amongst the coaches at my rink. Sometimes they are strongly opposed to each other's methods on certain things. It's an even bigger problem when my kids go to LTS where they have random coaches. Sometimes the LTS coach has said to them that they don't care what they're doing in their private lessons or even in a previous LTS class, they need to do it that particular coach's way in the LTS class. Then they won't be able to something well that they can normally do just fine, and won't progress through the LTS classes as quickly. It's worked out well enough though since there's different coaches all the time. It's relieving now that the older ones have passed all the Basic levels - I basically just wanted them to get past those so that they are welcome on freestyle sessions. Honestly LTS is a bit of a chaotic mess, haha. But it's good for working through all the basic skills and getting exposure to many different coaches.
It is likely that some of the readers of this forum are inexperienced coaches - including coaches who wish to lose fewer students to other coaches.
A thread like this one needn't just be useful to the o.p. and other students who consider switching. It could also be a lesson to such coaches on the value of maintaining good relations with everyone they can.
If students are aware that a coach resents other coaches, other coaches probably are too. Which means students and other coaches are less likely to recommend that coach. It could also influence decisions on whether one is allowed to teach.
And showing up promptly and reliably does matter.
I also think it worthwhile to elicit feedback from students on what and how one is teaching. That's an important part of adapting to students' goals and learning styles. Which, IMO, is one of the hallmarks of a great teacher of any subject - not just skating.
My last coach changed the schedule once without telling me - I was there for my lesson and he was coaching someone else when it was my time, ignoring me when I'd try to make eye contact and focusing on the student instead (I didn't want to outright interrupt). I was upset about it because I felt not taken seriously so I ended up taking a break for a while to just sit down, then get a text from him a half hour later asking where I am. When I talked to him he said he had to change the time to later but forgot to tell me. Other times he told me the day of the lesson, when seeing me at the rink, that he had something else to do so needed to cancel. If he did let me know he needed to cancel by text in advance, it was very shortly before the lesson. I understand needing to cancel sometimes but letting people know in advance if at all possible is preferred, certainly before the student has come to the rink expecting a lesson. I no longer have that coach and don't want to work with him again.
The resentfulness between coaches is frankly unprofessional. I don't always like all of my coworkers, but I'm not going to go talking bad about them or putting down their work behind their backs. It makes the coaching landscape difficult to navigate for students and parents. Of the many coaches at my rink, only a handful are willing to come to morning freestyle sessions when we have time to skate (or only will if we pay for a longer block of time than we'd prefer), many of those who are are already engaged with other students. We have to try to find ones that are available, can be reliable, provide worthwhile instruction, and have personalities that click with the students - having to worry about which coaches get along with other ones is an extra challenge we really don't need when paying a lot of money for the coaching time. Each student needs to have a primary coach whose specific instructions trump any secondary coaching, without resentment and contradictory statements from secondary coaches about that. Sometimes the most proven coaches are the worst about this, which is an unfortunate irony, since you both want to employ them thanks to their track record but at the same time don't want to because of the negative politics.
I haven't replaced my former coach yet, trying to take my time with making that decision. It's frustrating how difficult it is. It's hard to try out a coach and then leave them when problems come to light later - some coaches will blacklist skaters for that, and there's really only one rink that's viable for us to go to regularly. It's especially difficult when you have multiple skaters you're trying to coordinate and align coaching for.