The reality is plenty of the best jobs in most sectors (except perhaps local authorities and there pay is capped below the prime minster etc I think now) are for the privately educated because the education is better. Obviously lots of private pupils are not clever and wouldn't even then get jobs where you need brains like being a surgeon or judge, top accountant etc but plenty of them are.
A lot of jobs require:
(a) obviously the best exam results you can get
(b) social skills, ability to wear the right clothes for that job, get on with the clients, customers etc you will be meeting in that business
(c) being understood, speaking coherently and knowing where to put apostrophes, excellent written English, having an accent those with whom you will deal will understand
(d) confidence and to an extent reasonable looks/charm
(e) things in common with those with whom you will work
The biggest hurdle for many people is not the fact they are female but they don't have those things.
So how could the state system achieve them?
Obviously number 1 is getting as many A* as the private system
That might mean teachers working to 6pm and Saturday school and more homework perhaps supervised at school
Much longer school days with a wealth of clubs on offer which again involves teachers being prepared to stay until 6pm at least.
More selection and teaching in one ability groups.
Much more silent classrooms of teenagers. Some state schools manage this but not all.
Ability to exclude more easily with disruptive children going to the units they do to now but perhaps in greater numbers.
Making chidlren speak better. If my mother int he 1940s with classrooms of 40 children in a rough area could teach the children of the poor correct English state schools can now. Not necessarily just accent although loads of people change their accent to get on in life, but just basics - no "you was" I aint. Haitch etc. Teachers to tell children who they speak at home is not how they will speak at school.
If you listen to some groups of teenagers such as those on Jamie's Dream School one of the biggest hurdles they have is the inability to convey meaning because of their very very poor language skills.
Learning the cultural norms of the group with whom you will work whether that be opera, horses or bingo, depends on the group. Being able to relate to others is the point. Some people can go to an interview and make connections with that person and others can't. I think it's been one of my most useful skills. Whoever those people are I can connect - if they're Geordie that's where I'm from. If they have children I have mine. You need to learn how to make connections with people.
Private schools and probably good state schools teach children how to converse, debate, hold their own in arguments and activities like hiking until you're really tired, survival etc, ability to cope with very very long days which you m ay well need if you work in the City in an accountancy practice, ability n ot to be a jobsworth - downing tools but having an internal ability to cope whatever the conditions, that kind of survival skills comes in very handy in plenty of jobs and I suppose it is also something you might well learn on a tough council estaet too of course if you're up all night lookuing afetr your 5 small brothers because your mother is out for the count on heroin.
Probably plenty of other things too but I'm cooking lunch....