An example similar in spirit to yours is giving explicit examples of affine group schemes. Take, for example, $GL_n$: if we wanted to work solely in $\text{Aff} = \text{CRing}^{op}$ we'd have to write down a comultiplication and antipode on $\mathbb{Z}[x_{ij}][\det^{-1}]$, check the Hopf algebra axioms, etc. Instead we can argue using the Yoneda embedding, which is to say we can work with the functor of points $R \mapsto GL_n(R)$. Then we just have to check that $GL_n(R)$ is always a group and this group structure is natural in $R$. This boils down to the naturality of matrix multiplication as in your example.
A related funny example is the "affine ring scheme" $R \mapsto M_n(R)$, from which we can obtain $GL_n(R)$ as the group of units. Working solely in $\text{Aff}$ corresponds to writing down a commutative ring $\mathbb{Z}[x_{ij}]$ equipped with two comultiplications, one for the addition and one for the multiplication in $M_n(R)$, and checking the Hopf algebra axioms for one, the bialgebra axioms for the other, then some awful distributivity axiom between them. Meanwhile checking that $M_n(R)$ is a ring and that this ring structure is natural in $R$ is very straightforward.