Without answering this question on Hatcher's proof, I mention that the proof of the Cellular Approximation Theorem in Section 7.6 of my book "Topology and groupoids" uses no simplicial subdivisions, but only cubical ones, and you can find the same proof in older editions of this book, with different titles, which may be in your library. I took it long ago from handwritten notes of J.F. Adams. (The proof given restricts to the case of finite cell complexes, but the topology in the general case is arranged so that the same proof goes through!) Here is the key step.
The following statements are true for each $n \geqslant 1$.
$\alpha(n)$ Any map $S^r \to S^n$ with $r < n$ is
inessential.
$\beta(n)$ Any map $S^r \to S^n$ with $r < n$ extends
over $E^{r+1}$.
$\gamma(n)$ Let $B$ be path-connected and let $Q$ be formed by attaching a
finite number of $n$-cells to $B$. Then any map
$$ (E^r,S^{r-1}) \to (Q,B)$$
with $r<n$ is deformable into $B$.
The proof is by induction by means of the implications
$$ \gamma(n) \Rightarrow \alpha(n)
\Leftrightarrow \beta(n) \Rightarrow \gamma(n+1)$$
the only difficult step being the proof of $\beta(n) \Rightarrow
\gamma(n+1)$.