An article on the BBC News website failed to satisfactorily answer this question so I decided to have a go:
Eton has produced so many Prime Ministers because they teach a different curriculum to state schools. State schools’ purpose is not intellectual development but socialization in obedience and subordination.
Eton’s curriculum includes: literacy (public speaking and writing), study of law, literature, philosophy, theology, history, understanding the workings of government, independent work, physical sports, access to leaders, responsibility, personal code of standards, morality, an appreciation of music, design, literature, drama, art, observation and recording, ability to deal with challenges, a habit of caution in reasoning to conclusions and the constant development and testing of judgement.
These are not the skills required to fill the majority of jobs, such as shop assistants and nurses.