I think confidence is produced when you know you have access to or can have access to the best of various different key ingredients of success. SOme children from state schools will have this access or ability to access and that will boost confidence, so it's not just about schooling, but those in independent schools are more likely to be offered access by both school and home situation.
So confidence is boosted when you know you have access to great teaching, to great facilities, to great books and learning resources, to tutors or extra help if it's needed, to those who have inside knowledge about how to succeed and will share that info with you, access to people who can help you whether it's with GCSE choices or Uni applications, to a wide choice of sporting, musical and other activities with quality coaching provided, or getting work experience, or accessing a mortgage, or meeting some people who know about a particular firm or industry, or..... You basically know that the chances of failing are very low.
So, just knowing you have access to that stuff, and also that failing is therefore unlikely, or if some kind of failure happens,not here are lots of safety nets and protections in place to help you, makes a lot of difference. Fear is removed, because what is there really to fear if everything that could help you is in place or can be put in place.
So parents can put a lot of this stuff in place, regardless of schools attended, but some kind of knowledge and savvy is required, which all parents simply don't have. So some simply can't help their kids with their school work or give them expert advice about GCSE choices which open doors to the next stage in an academic way, or which Unis might be best for which industry, or supplying a tutor, or access to top sports coaches and musical opportunities. Independnet schools are very experienced in providing a lot of this stuff ....so their children are pushed towards certain types of course and offered a myriad of extra curricular options, so getting involved is just really easy and doesn't require lots of work, and when they find something they like, quality coaching is readily and easily available, plus experts to advise on uni applications, or contacts who can get them work experience whilst at school or contacts during uni years when thinking about the workplace.
So if through your whole life, you've always known that you WILL have access to or can have access if you need it, to the best and that there are lots of people looking out for you who know how to make you a success, it should really boost your condifwnce.
Conversely, if you look around you and know that what is available to you is somehow inferior to what lots of people are getting, then there is no surprise if confidence diminishes. If you look around and know that in your school there are lots of teacher shortages, or you know the school doesn't have books or you hear about other places which have super facilities that your school doesn't have, or clubs and activities that you've never even heard of, then you know things are not equal and some other people elsewhere are getting access to stuff you might not. If you don't have confidence that your school or your parents can provide you with good advice for the next stage of life, or introduce you to those who can, then you may well feel disadvantaged. If you know that if you struggle with someone, there probably won't be much help, or that if you take a risk and things go wrong, there won't be much of a safety net, then your willingness and ability to take risks might well be less.
Lots of children in lots of state schools have huge confidence and self belief. Usually some of it comes from their schools supporting and boosting them, but crucially there is also a home environment telling them they can achieve and providing resources, support and back up in all kinds of ways. Fear is removed and those children believe the future will be bright and they as individuals will have access to and deserve access to everything that is needed for success.
If you don't feel you have access to the stuff needed for success or that somehow you don't deserve access to it and that those things and success itself are things that belong to others but not you, confidence doesn't grow.
So yes, public debating and being taught to shake hands with adults might help.....but it's a much wider issue about having access to a wide range of resources and knowing you will have access to them and expecting to have access to them, which removes fear. The uncertainty about not knowing if they will be there, or suspecting they won't, whilst knowing others are having access diminishes confidence for most.
And yes, there are always some who despite circumstances have huge self belief and confidence....but they are smaller in number.
So it's not just about schools, although school type might provide access to those opportunities and supports which build a sense that failure just can never really happen to you, but schools together with home circumstance and expectations of life and access which makes the difference. That's in my view!