DH was privately educated, I was not, but went to the kind of outstanding state school that sounds like the alternative for your children. We both did competitive degrees at prestigious universities, both got first class degrees, both did postgrads at Oxbridge. He absolutely has that extra confidence people talk about, but it hasn't materially impacted on him. He has a network, which impacted more- he thinks his school was as much of an influence as his degree in getting him his first job in the city. (As an aside, I think those networks are still much more effective for men, but that's another story)
But, here is the problem- private education supports and equips you for a particular world: if your children don't necessarily want that world, it can actually be a negative. I work in the not-for-profit sector, I love it, and I made the choice early on. DH spent years in the city because he 'could' and because of the prestige- he is re-trained, and happier, in a much worse paid area which he probably always knew he wanted to go into but it felt like a 'waste'. I don't know if it's education or temperament, but I have enjoyed a lot more of my working life than him, and have had a lot less self-doubt.
The problem with the almost hereditary expectations of education you're describing is that it's ultimately self-defeating, unless you have a particular kind of child who turns into a particular kind of adult. I have a lovely friend who works in the city with the plan of retiring at 50 once the school fees are paid and then doing the job he really wants, the job he has always really wanted. That job, btw, has a pay scale of 40-70k, so it's not like his plan is to be a starving artist. I remember being 22 and talking to him, way before kids were on the horizon for either of us, and him saying how he would love to work in X but it would never enable him to send his children to private school and he felt it would be unfair to deny them the chances he got. That kills me: private education is supposed to give your children the opportunities to be whatever they want, but hidden in that is the expectation that whatever they want better be well paid enough to keep the cycle going.
I don't think that's a message you should send your children, and it seems by going back to work for no reason but to feed the school fee monster that is exactly the message you would be sending.