Le teorie vanno e vengono ma le formule restano.--G.C. Rota. (The theories may come and go but the formulas remain.)

Perhaps the Wiki on the Adams operation and "Formal groups, Witt vectors, and free probability" by Friedrich and McKay provide a quick intro to the connections the OP is questioning.

"Today the jargon is that of K-theory, yesterday it was that of categories and functors, and, the day before, group representations." All three jargons are used in the refs above (and those in my comments), serving to present different perspectives on, or even generalizations of, the basic, originally discovered relations among the symmetric functions. Territorial instincts may compel some camps to claim the superiority (and even priority) of their insights, or approach, which is probably what Rota decries even though he was certainly guilty of this same behavior.

(Read the introductory paragraph of "Alphabet Splitting" by Lascoux: ... meals were followed by long discussions about the comparative merits of algebraic structures, Gian Carlo for his part tirelessly asking me to repeat the definition of λ-rings that he copied each time in his black notebook with a new illustrative example.)

Added Nov. 8, 2019

I'm fairly convinced that Rota expressed exactly what he meant to express--that the identities/properties of the symmetric polynomials lie at the foundations of "these and several other attractive theories." See the refs and comments in the MO-Q "Canonical reference for Chern characteristic classes," in particular, "Characteristic classes and K-theory" by Randal-Williams, the linked Wikipedia article on Chern classes, the relevant sections in "Manifolds and Modular Functions" by Hirzebruch et al., and the Wikipedia article on the Splitting Principle.

Added 5/27/21:

Donald Knutson, in "$\lambda$-Rings and the Representation Theory of the Symmetric Group," states, "the notion of $\lambda$-ring is built upon the classical Fundamental Theorem of Symmetric Functions," that much of classical algebra is based on this theorem, and "the general definition (of a $\lambda$-ring) is somewhat complicated ... and will be best understood by first analyzing one manifestation of the ring Z, its appearance in the simplest example of K-theory." In addition, "the main technical tool (In proving the Fundamental Theorem) is the notion of $\lambda$-ring, first introduced by Grothendieck in 1956 ... in an algebraic-geometric context, and later used in group theory by Atiyah and Tall ... ."

Edit 8/23/2021:

From "Ten lessons I wish I had learned before I started teaching differential equations" by Rota:

I have always felt excited when telling the students that even though there is no formula for the general solution of a second order linear differential equation, there is nevertheless an explicit formula for the Wronskian of two solutions. The Wronskian allows one to find a second solution if one solution is known (by the way, this is a point on which you will find several beautiful examples in Boole’s text). ... every differential polynomial in the two solutions of a second order linear differential equation which is independent of the choice of a basis of solutions equals a polynomial in the Wronskian and in the coefficients of the differential equation (this is the differential equations analogue of the fundamental theorem on symmetric functions, but keep it quiet).

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