Other benefits are -
-getting used to playing/performing in front of someone. Depending on the instrument/situation, some children may not have the chance to do this in other ways. I avoided almost all possible situations where I had to, including all but a few exams. Then when the time came when I did want to (e.g., audition for ensemble) much later in life, I find that I am terrified and shaky and it all goes wrong. Exams are a gradual introduction to learning to play under pressure. While I still wouldn't have liked it as a child, I can see now that it would have been good for me.
-learning pieces to performance standard. When I wasn't doing exams, I would work at pieces and get them quite good, enough that my teacher would leave them and go on to new ones. But they were never really, really polished, the way you get them after loads of practice for a performance situation. I agree that practising the same pieces - and only those pieces - for months on end is awful and not the way to learn music, but at the same time, there is something that you get from going from 95% ready to 100% ready that you might otherwise miss, bits of extra polish/interpretation
-exposure to music you might not otherwise want to do. I loved Baroque music as a child. It was the only section I really liked out of the exam lists, and I'd have chosen to spend all my time on that. But by working through exam curricula (even without doing the exams), I had to learn pieces from other eras, other genres etc, and it was both good for my musical development (even when I still didn't like it much), and also helped me to learn and enjoy things that I really didn't think I would. I wouldn't ever have done it by choice, but when it's part of the exam, I just did.
-learning technique that can be very valuable for later playing. I didn't do exam work on one of my instruments, and I missed out on some very useful technique, and wish now that I had had to do it. I spent more time on other areas that I enjoyed more. Now that I try to play in ensembles, I can see where having spend a lot more time on arpeggios would have been extremely helpful. But by not doing exams, I could avoid doing as much of it as I should, and in all the combinations and permutations required of the exams, and to exam standard. I did some of it, but without that purpose, not really enough.
-motivation to keep going even through the hard bits. All the things I mentioned are absolutely possible to do without exams, but some children will find it harder to do e.g., all the technique without having a goal, even when they know they should. And it was the same for practising generally. Even though I enjoyed the instrument, enjoyed ensembles, there was still an element of not wanting to practice. So exams that have an immediate deadline, and a tangible sign of progress, and a purpose to work for, can be useful for some children.
I don't like the idea of just following the exam stuff, only doing 3 pieces and nothing else, and exams can be dangerous when it encourages teachers/pupils/parents to do that to the exclusion of everything else. But it's quite possible to do a whole swathe of material at each level, and then later on choose the favourites to work up for an exam, all whilst continuing to try other music and styles. Technique can be taught in lots of ways, studies, exercises and so on, and only learned in the proper exam format closer to the time. You can skip years and exams, or do the work but never do the exam at all. They aren't needed. But there are some good things about them too. Even though I would have dreaded them as a child, and wouldn't have wanted to be forced to do them. But with gentle encouragement and explanation and understanding, I might have been persuaded.