Hi OP. This is a really interesting thread!
I went to private school and so I know a lot of people who did. I’m also friends with many from state schools.
In my opinion... the key thing to remember is that a work ethic and intelligence, general characteristics are shaped by parenting. A state nor private education will ever trump what is taught at home. For this reason I witnessed bratty and uninterested kids at private school and I know clever and successful people from state schools. HOWEVER... generally speaking private school will have a far higher percentage of kids whose parents are invested - literally - in their success. This means there’s less disruption in classes, smaller classes, kids who ultimately are there to be polite to staff and teachers and it is a calm environment. From the things I have heard from some state school friends, their school experience was very different. I’d also say that private school does more than educate. There’s more time invested, teachers know the pupils better because they have the time. There’s a range of extra curricular stuff that, at the time I didn’t think much of, but I look back and realise I learnt a lot from it all. Bullying is minimal and if it is there it is taken seriously- again because they have the time to do it.
You can’t expect certain grades from a private school but you can expect your kids to reach their potential.
There’s a difference between boarding and just day private too. I found the boarders didn’t always have a strong grounding from parents which is perhaps where the abuse comes from against those who boarded. It’s a generalisation though. Parenting as I said above trumps any education.
As for whether to do private or house deposit... no matter what job you end up with, sadly it won’t magic 100k deposit. Is there no way to invest some of the money in the meantime - ie you won’t be spending 250k immediately? I know it’s not a helpful answer but I would try and do both. I loved school and I don’t know anyone who went to private school who didn’t - expect perhaps a few boarders. Even if you gift a smaller house deposit it’s still a start. Or maybe you could get a buy to let, if you have money spare at the moment, and use that income towards the school fees but keep the capital... another option would be to do state to age ten and you’ve already saved yourself circa 20/30k for a house deposit.