Discussions with family are probably going to have a different flavour from those with friends.
Family will be worried about whether you'll damage your child. IME, when it comes to home educating a primary-aged child, people tend to feel it's okay in the short term. The kinds of things an eight-year-old needs to learn are clearly not that hard to teach. Especially with you having a background in education, your relatives will feel fairly comfortable about that. Probably they will look further down the line: what if she gets lonely? what if she doesn't learn to get along with people? how will she do GCSEs? The simple way to reassure them is to address those questions somewhat - you may already have some ideas - while repeating that it doesn't have to be a permanent decision and she can always return to school later if home ed doesn't seem to be working out socially or educationally. "We're going to have a break from school and try a different approach for now. We'll see later what seems best at secondary age."
That line just kicks the can down the road, of course. However, since you have a hunch that your family will come round in time, that buys you a few years to let them see for themselves how home ed looks in practice. People who aren't familiar with home ed have trouble imagining it. Let them see. I bet that once you have been doing it for a few years, most of their worries will have evaporated. Also, you yourself will have learned more about home education and will have more answers for them!
Friends may well feel threatened by what you are doing. Most people acknowledge that there are some major disadvantages to school. Most kids don't particularly like school, and some are seriously unhappy there. When people send their kids to school despite those problems, this must be because in their minds there are even bigger benefits to school... or to express it differently, there must be big drawbacks to home education. If your home education goes brilliantly, that undermines their belief in those big drawbacks. (That's less true if your child's needs are obviously different from their own children's needs, which is why many people are comfortable with the idea of home educating kids who have special needs or who are being severely bullied, but believe HE should not be contemplated in "normal" circumstances.)
And if it isn't the case that there are big drawbacks to home education, it follows that maybe their own children are suffering unnecessarily at school, that maybe they themselves are letting their kids down. That would be a very painful idea. I think this is why some people get irrationally upset about home education.
The tactful way to skirt round all of that is to refrain from criticising school, and to hint at special circumstances for your family / your child which might mean home ed works for her but isn't necessarily best for theirs. ("She quite likes a quiet environment, and that's easier to provide at home." "Most of her friends aren't at her school anyway, so she'll be free to see them more often." "She has grandparents who are in ill health, and it will be nice for her to spend more time with them while she can." "She does better with hands-on learning." "She just really doesn't like school, so we thought we'd try something different.") The fact that your older children are staying at school will help with this.
If you actually do believe home ed provides better opportunities for most kids across the board, suggesting otherwise will feel disingenuous. However, it's rude to challenge the heartfelt beliefs of acquaintances who aren't champing at the bit for that discussion. Together with politics and religion, sometimes it's better not to go there.