From the first paragraph, where he explains the purpose of the article:
> The two fundamental facts about matrices is that every matrix represents some linear function, and every linear function is represented by a matrix. Therefore, there is in fact a one-to-one correspondence between matrices and linear functions. We’ll show that multiplying matrices corresponds to composing the functions that they represent.
And later:
> The connection is that matrices are representations of linear transformations, and you can figure out how to write the matrix down by seeing how it acts on a basis.
If there's a meaningful difference between "a matrix is a representation of a linear transformation" and "a matrix can be viewed as a representation of a linear transformation" it seems largely philosophical and, in any case, tangential to the author's stated goal of explaining why matrix "multiplication" is defined the way it is.
Whether or not matrix multiplication is a "particularly useful operation on a rectangular array in general" boils down to a debate about what is or isn't useful to do with a rectangular array. I'll leave that to other folks with stronger opinions on the matter.
And yes, understanding that is very important to motivate high school students. Affine transformations provide a good context for that motivation, as well as a good framework for intuiting noncommutativity of multiplication.