Alex, I hope you will allow for the option that there are natural explanations for the Chebyshev polynomials which are unrelated to optimization on $[-1,1]$. Here is what I think is the most natural characterizing feature of this family of polynomials.

Theorem (Ritt): If $F_n(x)$ is a family of monic polynomials with coefficients in a field of characteristic 0 such that $\deg F_n(x) = n$ and $F_m(F_n(x)) = F_n(F_m(x))$ for all $m$ and $n$, then up to a simple change of variables $F_n(x) = x^n$ for all $n$ or $F_n(x) = (1/2^{n-1})T_n(x)$ for all $n$.

Proof: See Ritt, "Prime and Composite Polynomials," Trans. Amer. Math. Soc. 23 (1922), 51--66.

Also see the chapter "Commuting Polynomials" in Kvant Selecta: Algebra and Analysis Volume 1.

The link provided in the original question mentions that the monic Chebyshev polnyomials commute, but doesn't emphasize that this is an incredibly special property of them. Pity.

Alex, I hope you will allow for the option that there are natural explanations for the Chebyshev polynomials which are unrelated to optimization on $[-1,1]$. Here is what I think is the most natural characterizing feature of this family of polynomials.

Theorem (Ritt): If $F_n(x)$ is a family of monic polynomials with coefficients in a field of characteristic 0 such that $\deg F_n(x) = n$ and $F_m(F_n(x)) = F_n(F_m(x))$ for all $m$ and $n$, then up to a simple change of variables $F_n(x) = x^n$ for all $n$ or $F_n(x) = (1/2^{n-1})T_n(x)$ for all $n$.

Proof: See Ritt, "Prime and Composite Polynomials," Trans. Amer. Math. Soc. 23 (1922), 51--66.

Also see the chapter "Commuting Polynomials" in Kvant Selecta: Algebra and Analysis Volume II.

The link provided in the original question mentions that the monic Chebyshev polnyomials commute, but doesn't emphasize that this is an incredibly special property of them. Pity.