Going back to OPs original question - yes, I think it is possible on that salary, though you might be cutting back on 10k holidays a year, unless in that you also include school trips. School trips are a pretty central part of school life, private or state, but at private there is so much on offer. My DC could have gone on two or three foreign trips each year from 1st form on, as well as CCF trips, field trips etc. These add up, along with sports kit, travel and uniform, to 22k per child easily each year and that cost rises noticeably year on year.
I agree you don't want to be scrimping for every penny, but we've made a few adjustments because we do have to be quite careful with money now, and tbh I barely notice them now. We spend less on groceries just by not overbuying. We spend a lot less on holidays by having our main holiday out of high season and just staying in UK for summer.
I can't think of any state schools, except some of the grammars) that offer equivalent education to the best private schools. I don;t agree that all private school children get their confidence from home. DS2 due to a number of significant physical and neural disabilities had zero confidence when he arrived at school. they picked up on this immediately and whisked him off to counsellors and SEN and groups and gave him a 6th form mentor. Within a year he was transformed. State schools rarely have the resources to do this. And I agree with Happy on bright children. Our local village school was indifferent to DS1s enthusiasm for learning. He's flourished at a school where the teachers are energetic because, by and large, the cohort is polite, works hard, is bright and aims high, so they aren't expending energy on crowd control.