State schools have "much to learn" from the private sector, Dr Anthony Seldon, master of Wellington College until this year, told a conference on Friday.
Their "remorseless drive... for exam success is no longer fit for purpose", he said at Tatler's Schools Live.
People need the skills to do things that computers cannot, he added.
Dr Seldon, the first speaker at the event, explained why he believes pupils need to learn teamwork, empathy and resilience to be ready for life beyond the classroom.
In his speech at the conference, which is aimed at parents looking for information on independent schools, he said: "Independent schools are taking the lead nationally in preparing students for the jobs required for the 21st Century."
Dr Seldon, now vice-chancellor of Buckingham University, quoted a recent Harvard University study which found employers need far more than the skills developed in exams.
"They also need what is patronisingly called the 'soft' skills, i.e. those that cannot be replicated by computers, which are fast taking over not just manual but professional jobs also.
"These are the skills of creativity, teamwork, empathy, grit, resilience and honesty."
Dr Seldon said England's education secretary, Nicky Morgan, is the first "fully to appreciate" schools can excel both in academic rigour and at teaching character.
"The reason why alumni from independent schools are so dominant across society is not just because of the excellent exam results they receive, but precisely because of the grounding in the soft skills.
"I am expressly not critical of state schools themselves. They are the victim of forces that compel them to focus on a narrow range of exam teaching and subjects at the cost of broader education in the arts, character, sports and the social and work skills that employers increasingly want in the 21st century.
"Some state schools manage to do exams well and offer this breadth of education, but it is much, much harder for them than independent schools," Dr Seldon concluded.
It is not good grades but a "grounding in soft skills" that gives people who went to independent schools their edge, a former public school head has argued.