Longest good question about soft skills. If you are wanting to supplement state school activities here are things that have done that in my experience
Sport - especially team stuff because of things like the etiquette (clapping the other team into the clubhouse, shaking hands), awards evenings, getting on with team mates and overcoming differences of opinion etc
Music - orchestral music playing can really broaden education and give cultural confidence. Like sport there are high expectations of behaviour, lots of self discipline needed to deal with waiting around while another section works on a problem, confidence from starting a new piece and being rubbish and then improving through hours of hard slog before playing something really well.
Drama/singing/musical theatre - being able to speak in public, do presentations, just act really. Cultural capital again.
Of course the above requires major parental investment in terms of time, money and energy. You could just choose a good independent school who will do all this for you 
I've had DC at both. My DC are bright and dyslexic so not the "bright kids will do well anywhere" kids. It is hard for state schools to offer much support to kids that will easily get 5 GCSEs as their support staff is being cut to the bone.
In my experience the areas where the Indy beats the nice state schools we have used are:
Choral singing - (hard to arrange outside school where I live, nearly impossible in fact). DC at private school sings three times a week.
Exercise - again 3 or 4 times a week and loads of variety of sports, high standards of coaching, facilities. Even if they're not in A teams they are out excercising. DC not a lover of team sport but is fit.
Food - hugely better than state offerings. DC is hungry after school but not starving like they were at primary and like sibling at state secondary is. There is a great choice of hot meals, salads, veg, side dishes, all served buffet style so they can eat plenty. Only one serving of pudding allowed. Lunch time has well managed sittings so everyone gets chance to eat. All the staff eat in the dining room and atmosphere is nice.
Staff more able to support, advise, offer help. In DT if a student wants to test something out they will set up an experiment in class or if it is big in the lunchtimes.
Curriculum - staff can teach what they want early on so can incorporate really interesting stuff.
Lots of homework could be a downside but it is always related to what is being done in class so DC knows what to do and does not need a lot of help.