"I suppose that the advantage with school is that you do get to see the same friends every day. "
Depends on the set up. Some HE families do HE in groups, or have regular, even daily activities with the same families.
We don't see the same families daily but probably see the same 3 or so families 3 times a week, plus on top of that activities and groups where we see a wider mix of people.
I think what you are saying is that you want to take her out but she is worried about the social side.
If so-well she probably hasn't got much of a conception of how home education would work, most people haven't. Why not get in touch with your local HE groups (pretty easy to find but shout on here if you have any problems) and take her along to a few meetings (a few is best, they can be overwhelming at first!).
It seems quite reasonable to me that she would want to go to school if her friends are there, actually. I once worried about leaving work for much the same reason. I am not sure that I'd let a 9 year old make the decision entirely though, if you honestly feel that it would be in her overall best interests to take her out, then I would do so.
Actually in your exact situation I'd consider flexo-schooling, which is where she is in school for a few hours a day/week. Then she'd get the social side and you'd get to teach her at home. You do have to follow the NC but aside from the fact that its the sort of stuff that even a family that didn't teach would be doing anyway, it sounds as though you teach her, so could presumably follow the NC without a philosophical meltdown.