The proof of the Dold-Kan theorem basically amounts to the following. Let $\mathbb{Z}\Delta$ denote the pre-additive category generated by the simplicial indexing category, so that $s\mathrm{Ab}=\mathrm{Fun}^{\mathrm{add}}(\mathbb{Z}\Delta^\mathrm{op}, \mathrm{Ab})$, the category of additive functors.
Let $\mathcal{C}$ be the "Karoubi envelope" of $\mathbb{Z}\Delta$, the category obtained by freely adjoining splittings of idempotents. You can identify the Karoubi envelope with a full subcategory $s\mathrm{Ab}$, namely the closure of the image of the Yoneda embedding under splitting idempotents.
Then it is trivial that $\mathrm{Fun}^{\mathrm{add}}(\mathcal{C}^\mathrm{op},\mathrm{Ab}) \approx \mathrm{Fun}^{\mathrm{add}}(\mathbb{Z}\Delta^\mathrm{op}, \mathrm{Ab})=s\mathrm{Ab}$. The Dold-Kan theorem amounts to observing that every object in $\mathcal{C}$ is a direct sum of objects $G(n)$, and that the full subcategory of $G(n)$s is the "indexing category" for chain complexes. You also have that $G(n)\approx \mathbb{Z}\Delta[n]/\mathbb{Z}\Lambda^0[n]$.
The usual formulas for normalized chains arise by explicitly identifying $G(n)$ with either a (split) subobject or a (split) quotient object of $\mathbb{Z}\Delta[n]$. In fact, there is only one way (up to sign) to identify $G(n)$ as a split subobject of $\mathbb{Z}\Delta[n]$, and this translates into the "quotient by degeneracies" construction of normalized chains. There are several ways to identify $G(n)$ as a split quotient: the usual presentation of normalized chains as subobjects is induced by the projection $\Delta[n]\to\Lambda^0[n]$.
Of course, you could also be perverse: $G(n)$ is a summand of $\mathbb{Z}\Delta[k]$ for any $k\geq n$, so you could define normalized chains using your favorite idenification of $G(n)$ as a summand of $\mathbb{Z}\Delta[n+42]$, if you want!