I've been in your instructor's position a few times, and I've threatened to demote. But I recognize that taking away a belt can be detrimental in several ways (call me a snowflake if you want).
First, a kid works hard for the belt. Then you take it away (or prevent him from wearing it). That's embarrassing. And what have you done? You've let a single incident (of typical behavior for that age) take precedence over a few years of otherwise hard work. Bottom line: not really that fair. You focused on the negative, and no so much on the positive. I know the counter argument: I didn't really take it away - the kid caused it to be taken away. But that's a nuance lost on 12 year olds, I think.
Second, you run the real prospect of losing that student, who will then turn around and bad-mouth you to his friends. Then you have a reputation to deal with. For some instructors, not a big deal, especially in a large town or city. But for others, that reaction is cause for reconsideration, if it causes you loss in business. I don't teach for a living, so, they can bad-mouth me all they want, it's not going to change me.
Third, what have you taught? I mean really: what have you taught? Say you taught them a form/kata. Now you demote them. Think about what the belt really means. To them, it's a marker of achievement. But to you, it's a means to divide up a class: White belts in that corner, green belts in this corner, and red belts go over there. You've just made your teaching structure that much harder by removing your tool to measure who needs to learn what. Maybe you can deal with a single student with the off-belt (do you really have dozens of students with demotions, even in the largest schools I think probably not. LOL).
Fourth, who are these kids? In my case, I have many a student who, shall we say, "don't fit in a box". They have issues. Because of the environment my school is in, most have behavioral issues. Others come from broken families. Others have issues only they would know. For better or worse, my classes are my classes. That is the hand I am dealt. I can kick them out, expel them, or I can suspend them. Or I could make them a better person for it.
For me, I have a rigid set of standards of behavior I expect of all my students and instructors. But they are never threatened with expulsion (except in obvious egregious cases, which I've never had to do). I have special tasks they have to volunteer to do, such as cleaning the dojang floor, putting away the equipment, washing down the mirrors. Those with behavior issues are "voluntold" ahead of those who volunteer. In one case, I had a student mouth off to one of my assistants. His "punishment" was to come to classes early or stay late to read a book. During class, he participated as if nothing happened. When he was done with the book, he and his mother came to class, and he had to explain his behavior, and why he read the book. He also had to explain what he read. Of course, he had a sit-down with the assistant and me, and we discussed why it's wrong to mouth off the way he did. Sometimes, the instructor is wrong, or doesn't know how best to deal with emotional-needs students.
I have kids in class who constantly use colorful vernacular to each other. I accept that it is cultural, and not of my own culture. But if I demoted or expelled every kid who used the n-word, I wouldn't have a class at all. Indeed, for some I also have their parents as students: do you think I'm going to teach an old dog new tricks? Instead, I make it a point to teach them about respect for each other, and to be considerate of those who may overhear. This is a constant battle, and therefore, is a significant part of my teaching. In fact, it's a subject I begin and end nearly every single class. Usually, I begin each class with a short dialog on respect for others, hoping that will rub off during class. And I end a class with a short dialog on respect for themselves, hoping that will rub off at home.
Can you demote? Sure. But before you go down that road, remember there are several stakeholders in that student. There is the student of course, but there's also the (paying) parent. By working with the parents, you can safeguard against the quitter, who ultimately is the loser because he didn't get a chance to reflect, to learn, and to improve. That quitting student might have quit at the behest of the parents, who didn't feel the need to pay for a student to be demoted.
In the end, I think positive discipline begets more respect in the long run. It may take awhile for some. I have many kids with many issues, and they all know each other and each sees each other making mistakes. Dumb mistakes. But they also see that I try to help them, guide them, and mentor them. I respect them, I do not bully them or judge them. And I accept them. By demoting them, that (to me) is demeaning. I would rather be more constructive.
So I said all that to say this to answer your question. Is it ever appropriate? In my book, no. I think there are always better methods. I think that everyone loses in a demotion, and no one learns. IMHO, the teachable moments are much more desirable rather than the punitive moments.