The issue comes from misunderstanding this sentence:
Today it is K-theory yesterday it was categories and functors, and the day before, group representations.
The listed objects are not a list of theories. If they were, he would say "category theory" and "representation theory". Instead, it is a list of different languages, or as Rota calls them, jargons. The sentence with the list is connected to the previous sentence, explaining what jargons he is thinking of, and not so much the sentence after.
We could remove the sentence, making the meaning clearer:
In all mathematics, it would be hard to find a more blatant instance of this regrettable state of affairs than the theory of symmetric functions. Each generation rediscovers them and presents them in the latest jargon. [...] Behind these and several other attractive theories stands one immutable source: the ordinary, crude definition of the symmetric functions and the identities they satisfy.
The "theories" in question are the theory of symmetric functions, rediscovered in different forms. For instance presumably one of them is the character theory of $GL_n$, expressed in the language of group representations.