This is perhaps unanswerable, or perhaps I am too algebraically ignorant to phrase it cogently, but:

Is there some identifiable reason that polynomials over $\mathbb{C}$, $\mathbb{R}$, $\mathbb{Q}$, $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$ are so pervasively useful in mathematics?

Is it because polynomials are in some sense the most natural functions defined on a field? I know that every function over a finite field $\mathbb{F}^n \to \mathbb{F}$ is a polynomial. And, by the Stone–Weierstrass theorem, every continuous function on an interval can be approximated by a polynomial. Is this the universal aspect of polynomials that "explains" their ubiquity?

Even tropical polynomials, which employ alternative addition/multiplication operations forming a semiring, are proving useful.

I'd appreciate your insights!