Consider the proof that for a projective variety $Y$, the regular functions are only $k$. $S(Y)$ denotes the coordinate ring, and $S(Y)_N$ the homogenous of degree $N$ in it.
Hartshorne establishes first that given a regular function $f$, there is a large $N$ so that $f \cdot S(Y)_N \subset S(Y)_N$.
Given this, and that $S(Y)_N$ is a finite dimensional vector space over $k$, doesn't this imply $f$ is algebraic over $k$ directly which wins (i.e Cayley Hamilton).
He seems to pass through it being integral over the module $S(Y)_N$ and I don't see why you need to.
Your argument seems correct to me. I would guess that Hartshorne's motivation for picking the explanation he did is that it's more ring-theoretic, which he thinks would be more familiar/useful to the reader. (Just a guess, though!)