I think Abdelmalek Abdessalem and William Stagner are completely correct in their inteprtation of the words "Behind" and "one immutable source" as describing one theory, the theory of symmetric functions, being the central core of another.
The issue that led to this question instead 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 function of this sentence is to explain what jargons he is referring to in the previous sentence.
If we delete it, the paragraph still makes perfect sense, but lacks detail:
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 not K-theory, category theory, and representation theory but rather the theory of symmetric functions expressed in the languages of K-theory, category theory, and representation theory. For instance presumably one of them is the character theory of $GL_n$, expressed in the language of group representations.
The reason I am confident in this interpretation is nothing to do with grammar but rather the meaning and flow of the text. The claim that symmetric function theory is the source of three major branches of mathematics seems wrong, but if correct, it would be very bizarre to introduce it in this way, slipped in the end of a paragraph making a seemingly less shocking point, and then immediately dropped (unless the quote was truncated?). One would either lead with it, or build up to it, and in either case then provide at least some amount of explanation.
Thus instead I (and Joel, and Vladimir) interpret it as making a less dramatic claim.