Why do you assume she was arrogant and stealth boasting and over-confident?
I was an extremely shy, anxious, underconfident perfectionist. I was also very well-behaved and quiet, and might well have been picked to do a tour, which I would have hated doing. But I would have, and would probably have been a bit awkward about it, whilst coming across as cold and aloof and maybe a bit superior as a result. I was also at a state school.
I would definitely have felt that I'd not worked hard enough if I'd only got an A, when everyone knew I was capable of getting, and expected I'd get, an A, and I might well have said something like that, too, in a way to show that I knew I wasn't anything special, that I hadn't got the A, wasn't perfect, etc. It would have been the complete opposite of what you'd assumed. Kind of like getting the criticism of myself in before anyone else does.
Yes, to people who are used to Bs, it might have seemed arrogant or faux modesty or something, but it really wouldn't have been. When you are in an environment when you perform highly and others do, there is lots of genuine discussion around A or A* or whatever, that has nothing whatsoever to do with boasting. It's just that you are talking about what is relevant to you. Just like in a situation with a music school crowd, who are discussing merits of various grade 8 pieces at a young age, because to them it's exactly what is relevant, and nobody thinks it's any kind of boasting, because that's what matters to them. To anyone listening who isn't part of that environment, it might well sound like stealth boasting, but it doesn't mean it is. And yes, they might sound precocious or arrogant, but that's partly because they're children and haven't learned yet how things sometimes come across. Or sometimes there is an element of arrogance that they haven't learned to tone done. But very often the whole thing is in the mind of the listener. I hate hearing things referred to as "stealth boasting" on MN, when often I think it really is just normal life for the people who are discussing something, and that it's insecurity on the part of a listener who assumes that it must be stealth boasting, done somehow to make them feel worse.
If you don't like the feeling you get from a school, fine. But judge it based on more than one comment like this. And ditch the whole "we're working class and we weren't brought up to boast like that" attitude, because it's nothing to do with class. If you find that the school generally seems to produce a culture of arrogance, then that's a perfectly good reason not to choose it. I wouldn't like it either. But I think it's a characteristic of some schools and not others (yes, even within private schools), and independent from class.