Perhaps the Wiki on the [Adams operation][1] and "[Formal groups, Witt vectors, and free probability][2]" by Friedrich and McKay  provide a quick intro to these connections.
	 	
"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 compell 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][3]" 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 Arnold 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][4]," 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.


  [1]: https://en.m.wikipedia.org/wiki/Adams_operation
  [2]: https://arxiv.org/abs/1204.6522
  [3]: https://link.springer.com/chapter/10.1007/978-88-470-2107-5_17
  [4]: https://mathoverflow.net/questions/345437/canonical-reference-for-chern-characteristic-classes