I think you are misinterpreting the quote. In the last sentence, the word "source" does not mean "source of these theories (K-theory, categories, group representations", but "source of the theory of symmetric function". Rota is not claiming that K-theory, etc have "at their  core the ordinary, crude, definitions of symmetric functions", but that "the theory of symmetric function", tautologically, does. What he is saying, it seems is that the beautiful and rich theory of symmetric functions can and has been developed without the need of modern fashionable abstract theories.