My two sons were day students at Bootham from age 11 to 18, and like you, we looked into all of the independent (and many state) schools locally before making our decision. All I can say is they had a wonderful time, made a lot of friends, achieved excellent grades and have become caring, thoughtful members of society.
Bootham is indeed an oddity. It seems relaxed when you look around, no uniform to speak of, disorganised, lots of noise. But when you are a parent and see the school every day, you realise that there's something going on underneath. They don't need a uniform to bring them together or to enforce a discipline. If you sit and watch how they interact, they seem to have a mutual respect amongst themselves, and between themselves and the teachers. Everyone is polite. Everyone is happy. They actually seem to care for each other. From this respect comes results, because everyone is encouraged, no one is made to feel out of place. Everyone has a talent. So the students seem to achieve amazing things, whatever it is they're good at, hence I guess the high league table results. But results oddly seem almost secondary to the fact that these are children who are learning life skills as well.
I have since met the new head, Jonathan Taylor, and he is even more approachable than the previous head. The Quaker ethos is not shoved in your face - all it seems to do is instill a mutual respect and acceptance amongst everyone, so that can't be a bad idea for society as a whole! There is a daily 9am meeting, but everyone just sits in silence for 15 minutes and contemplates what's important to them. Nothing religious.
You indeed have a big decision to make. But, as someone who has been through the process, personally, I think the school is wonderful and the staff are amazing. I have always wished I could have gone there myself! As a product of an old fashioned, strict, grammar school, I never enjoyed my time at school, and I have always felt I could have learnt more if I had only felt motivated and happier. Hence the choice of Bootham.
As a final aside, I always remember that if I ever visited Bootham at lunchtime, I would be welcome to sit amongst them and have my lunch there too, and I would notice that the staff and students eat together, side by side, OUT OF CHOICE. That tells you everything.