Either you temper your behavior as a coach or you accept the fact that only skaters can tolerate your behavior will choose you as a coach. Perhaps that's what we're seeing the end result of in this topic and assuming that the yelling coach makes the competitive skaters better at skating.
I will chase the skater and call out things loudly, like "turn the shoulders," "PUSH," "You're not turning enough," "You didn't PUSH!" lol. But I won't yell "Are you crazy?" or "Move your ***" as some coaches do. I don't think you discipline through embarrassment, although it's almost unavoidable in synchro or groups.
For me, it works better to stop the skater and address the issue quietly, one-on-one if the reminders aren't working, as my example shows.
I have several students on a synchro team, which I do not coach. Their coach last year was a college student who I thought was really sweet and nice. I couldn't believe the stories the girls told me after the season was over, about the coach mocking them, yelling at them, and berating them. I actually said "Nah, she was just trying to be heard above the other teams' music."
One of the parents told me that the girls weren't embellishing - they had listened several times after their kids complained and the coach was not only loud, but mean! Rather than say something to the coaches, they instead told their kids to just brush it off because (again, life lessons!) they felt it was important for the kids to learn how to deal with someone they didn't like who was in a position of authority. As one mom told me, "There will always be a teacher, a coach, or a boss that you won't like, but you have to accept and deal with." I learned a life lesson that day and my own kids have benefitted from their wisdom. I have THE BEST skating parents and students. But, they're recreational skaters, not the competitive skaters that the OP is admiring.
Raising your voice is always a cultural thing but it's also a learned behavior. I have a friend whose coach was a yeller and she (as a coach) is a yeller, but her family is so quiet, it's out of character. My coach was German and is still yelling after all these years. That's how her mother coached, that's how she coaches. Her father's quiet, but everyone else is loud, loud, loud. I fit right in! I don't think the yelling does anything for her coaching, but I always felt comfortable yelling back, lol. (When she was a Director, I taught groups for her one season. I told her to stop yelling at the instructors in front of the students because it undermined our authority. She hadn't even thought of that - she comes from a loud family.)
I think that the skaters who have the attitude to reach the top of the heap can deal with the yelling and the brow-beating. It's a chicken-and-egg scenario: which comes first? Is the yelling coach really the most talented coach or is s/he starting with skaters that have talent?