OP, I am absolutely evangelical about the brilliance of boarding schools (and this is choice, rather than circumstance, as it is in your case). But I would absolutely not be if my DC were at a school which called them idiots. I would be removing them like the proverbial shot.
I am not going to pass any negative comment on boarding per se. But to respond to your OP...
There are mixed messages here. Your DD sounds unhappy at school. But she is also not happy for you to step in. And she has friends. She is only 10, and she is going to feel ambivalent about this.
You love her to bits and would presumably not send her to board in an ideal world. But you have a decent career, which presumably matters to you. So you are presumably also ambivalent.
It's so hard to know quite what your question is, and how people could advise you. I am a SAHM, so if my DC didn't 100 percent like boarding, I'd have them at home like a shot. Or after they had given it a decent chance, at any rate. When DC1 went at 13, I said we would give it a term, and then re-think if he wasn't happy by Christmas (I thought he would hate it at first, as it was so different from being at home). I think that having this as a security blanket might have helped, but he is now worrying about having to leave after he finishes his A levels, because he loves it so much
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Overall, the experience has been wholly positive for us.
OP, your DD is that bit younger, which makes it more difficult to gauge exactly how she feels and why. Nobody should be calling her an idiot. I'm not saying you should change career or your working pattern, but would it be possible, if you needed to? How receptive is your DD's school to parental concerns? A good boarding school will go above and beyond the call of duty to look after the children in their care. IME, they are a damn sight better at this than day schools are...