I don't think schools can be expected to provide everything. The family setting is important too.
My parents were/are cultured people and I grew up going to art galleries, reading literature, playing and reading about music. I didn't do GSCE or A level Art or History of Art, but I had Ernest Gombrich's book and several beautiful books on various artists and we went to art galleries as frequently as possible considering we lived in the sticks!
I did history at school but this was mainly 18th and 19th and 20th. I didn't study 20th century history but I had books on the two world wars and also books about the cold war so knew all about the Cuban Missile Crisis, the foreign policy of the US, Vietnam, Cambodia, Nicaragua. I read my Dad's politics books and I knew roughly who the PMs and governments were of the 20th century and stuff about the development of the labour movement and so on.
I didn't do philosophy GCSE or A level but I had an introduction to philosophy and could have told you about some of the main philosophers and what they wrote.
I didn't do classics but I had greek and roman myths and legends from when I was very young and then when older I had shortened versions of the Odyssey and the Illiad.
I didn't study GCSE or A level music but I had books on the history of music and could have given you a mini presentation on it from memory. I played two instruments well and did grade 5 theory of music. I knew about sonata form, and quite a lot of music theory.
I read Dickens, Austen, Eliot, Hardy and all the great 19th century novelists, together with Orwell, HG Wells, Evelyn Waugh.
This stuff was all lying around at home or was bought for me by my parents following me expressing an interest. It had little to do with what I was taught at school.
Not trying to show off or make myself out to be some kind of genius but this thread has really made me realise how much of my education didn't happen at school at all!