Teachers are a very diverse bunch of humans, and so are their pupils. It's quite possible to hunt around and find particular combinations of teacher and pupil where the teacher isn't right for the pupil - where their methods and attitudes won't get the best out of the pupil - but I'm not sure this is a particularly useful thing to do. It's perhaps better to accept that most teachers make a personal choice on whether to use exams, and how to use them. Every teacher I've ever met makes the decision with the pupil (and their parents if the pupil is a child), based on what all parties think will work best for that individual pupil.
There are undoubtedly teachers who have written that they wouldn't enter a pupil for an exam because they consider them too young. I suspect they're often thinking of specific cases. Many teachers take the attitude that learning music is not a race. It is a journey. The message behind "I will not enter them because they are too young" is actually "I think they would benefit from time to reflect, explore aspects of music theory and practice that are off the syllabus, enlarge as humans and musicians, and they really don't need this exam at this precise moment, so let's concentrate on making them a better, more rounded musician". That's not such a bad message, surely?
Serious thought: I do rather worry about an emphasis on exceptional pupils. Very, very little good comes out of elitism in any branch of human endeavor. When you employ a plumber, you don't care if he's the best plumber in the country, able to fit a tap faster than any other. You want someone who will fit a tap to a professional standard, that's all. It's better to have lots of competent plumbers than one plumbing genius. Even in sport, with its competitive focus, the elite ones are pretty to look at, but the health benefits, both physical and mental, only happen for people who pursue the sport, and the benefits are there even if you're rubbish at it. Everyone who plays tennis gets the work-out of a good tennis game, and the mental satisfaction that comes with their sport, not just Federer. Elite sport is where the money is, but accessible sport-for-all is where the benefits happen. The value of the elite is when the rest of us get excited and try to emulate them, but even that is a double-edged sword because it's all too easy to give up because you find you're "not very good". In music, only a small number of people can win competitions and festivals. The world doesn't need many professional concert pianists. But every human who plays or sings is enriched by it, and enriches the world around them. The teacher who brings music to the lives of average pupils is, I believe, far, far more useful to the world than the teacher who miraculously fosters genius. In fact, if you get enough people doing something reasonably well, a few geniuses will pop out of the pool automatically.
So that's where exams are not a waste of time. Some people hate them (I do). But for others they are very motivating. Some people want a measuring-stick to measure their progress. The majority of those who aim to win a major competition, will, by definition, fail, no matter how talented they are, because there's only one winner. It's even worse if you want to be a world-leader; you've chosen a measuring stick that elongates in response to how good everyone else is getting, in order to doom the majority to failure. If your aim is to pass grade 8, then ABRSM really don't care how many people pass, provided they show they've reached the level. The more musicians reach that level and pass that exam, the happier the board will be! And the better the world will be, too. Although equally, it doesn't matter to the world how many people learn to play and reach that standard without taking the exam.
At their best, exams are a way in which a wide range of people can measure their achievements and feel motivated that they're getting somewhere.