OP here! I wanted to check back in and say a very big thank you to everyone who has taken the time to share their experiences with me. It has been really interesting to read so many different viewpoints, and has really helped me to clarify my own thoughts on what we would be hoping to get out of a private education, if we were to go down that route.
I think where we are at is that we are going to stay in a bit of a holding pattern for the next year and consider a possible move in the year after - so for DS that would mean a transfer into Y9 (when there is natural movement into the school) and a Y7 start for the next DC. That gives us more time to research the options, experiment with living on the kind of budget that would be required, and see how our stable our employment is looking post-covid (it is still too early to tell for me). It also gives us more time to evaluate the current school.
I think @altopredominant sums up pretty well the concerns I have about the state school and what we would hope to achieve by switching. It isn't really about the academics, which as many people have said can be addressed in other ways, such as tutoring. It is about a richness of experiences and opportunities which just aren't available at the local state schools, and - if the school is the right one - about the simple day to day enjoyment of their school years.
I can't emphasis enough how limited our ability is to offer these things ourselves outside of school. I might not be having this discussion if we lived in a vibrant large town or a city (not an option, by the way). Just as an example, a PP mentioned that their non-'typical''sporty DC enjoying climbing - I really think that would be a great option for my DC, but the nearest climbing centre is nearly an hour away. The private school, on the other hand, has their own climbing wall and club. There is very, very little on offer locally unless you want to play rugby or cricket. Also, we have three children, and we both work and commute. There simply isn't the time in the day to support three very different children in a diverse range of out of school activities, especially when they involve lengthy drives.
What this thread has really shown me is that what I actually need is a crystal ball! There are so many different outcomes here, both good and bad, for both options. I suppose the best outcome is that all three children come to thrive in the local school, leave as rounded and happy young people to start the next stage of their lives, and we congratulate ourselves on having an extra half a million quid to retire on, or to help them on the housing ladder. But if one or more of the children leaves school as an unhappy young adult who had a pretty miserable time of it and has failed to reach their potential, it is going to really take the shine off those few extra years of retirement.
Alternatively - we move the children to private schooling and they take full advantage of every opportunity given to them and discover interests and passions they never new they had. We stay healthy and employed and actually enjoy working until we are 65, and look back on it as the best money we ever spent. The flipside - one or more of the children fail to flourish at the school, and is unhappy and feels like a fish out of water. One of us loses our job or becomes ill, and we end up stretching ourselves thinner and thinner and become sick with worry about the financial treadmill we are stuck on.
So there is no right or wrong decision, and all we can do is use the next year to gather more information that will hopefully steer us towards making the right one. Thank you all again for sharing your experiences with me.