Does homotopy invariance of homology follow from the structure of the simplex category $\Delta$? Explicitly: Let $\Delta$ denote the simplex category, and $\mathscr{C}$ any small category, and fix a functor $F:\Delta \rightarrow \mathscr{C}$ such that $F\Delta^0$ is terminal. Also, assume $\mathscr{C}$ has products.
Using $F$, we can define a homology theory in $\mathscr{C}$ letting $C^n(X) = \mathbb{Z}(Hom(F\Delta^n,X))$, the free abelian group generated by morphisms from $F\Delta^n$ to $X$. The boundary map is constructed from the face maps in $\Delta$, and one can prove that $\partial^2 = 0$- this is exactly what is done to find singular homology of topological spaces.
Now, suppose $f,g:X \rightarrow Y$ are maps in $\mathscr{C}$. We say they are homotopy equivalent if there is a map $h:F\Delta^1 \times X \rightarrow Y$ such that $h_0 = f$ and $h_1 = g$. Here $h_0 = X \rightarrow F\Delta^0 \times X \stackrel{F\delta^0 \times 1_X}{\longrightarrow} F\Delta^1 \times X \stackrel{h}{\rightarrow} Y$, where $\delta^0:\Delta^0 \rightarrow \Delta^1$ is a face map. $h_1$ is defined similarly.
My question is: If $f$ and $g$ are homotopic, does it follow that the induced maps of chain complexes $C^{\bullet}(X) \rightarrow C^{\bullet}(Y)$ are homotopic as well?
 A: Chris's comment suggests that very little about the target category $C$ is being used in the standard argument, but I still think there's something interesting to check, namely what exactly is being used. The question concerns first of all a "singular chains" functor
$$F_{\bullet}: X \mapsto \left( \Delta^n \mapsto \mathbb{Z}[\text{Hom}(F \Delta^n, X)] \right)$$ 
from $C$ to the category $\widehat{\Delta}$ of presheaves of abelian groups on $\Delta$ (equivalently, of simplicial abelian groups). After taking singular chains, the rest of the argument will take place almost entirely in $\widehat{\Delta}$. 
The question concerns second of all a notion of homotopy in $C$ coming from taking $F \Delta^1$ to be the interval object. The only relevant data in such a homotopy is the induced map
$$F_{\bullet}(F \Delta^1) \times F_{\bullet}(X) \to F_{\bullet}(Y)$$
in $\widehat{\Delta}$. This is the only place where we use that $C$ has finite products and that $F_{\bullet}$, by construction, preserves them. This is also the only place where we use that $F_{\bullet}(X)$ and $F_{\bullet}(Y)$ have anything to do with $C$. Next, note that for every $n$ we have natural maps
$$\Delta^n \to F_{\bullet}(F \Delta^n)$$ 
(where $\Delta^n$ denotes the corresponding representable presheaf of abelian groups on $\mathbb{Z}[\Delta]$) which arise as follows. By definition, a map $\Delta^n \to F_{\bullet}(F \Delta^n)$ is an element of $\mathbb{Z}[\text{Hom}(F\Delta^n, F\Delta^n)]$. But there is a distinguished such element, namely $\text{id}_{F \Delta^n}$. These maps in fact organize themselves into a natural transformation from the Yoneda embedding $\Delta \to \widehat{\Delta}$ to $F_{\bullet}(F(-))$, which is (a restriction of) the unit map of an adjunction. This implies in particular that they respect the face maps $\Delta^0 \to \Delta^1$.
Now it's clear that pulling back along these natural maps gives a natural transformation from homotopies involving $F_{\bullet}(F \Delta^1)$ to simplicial homotopies
$$\Delta^1 \times F_{\bullet}(X) \to F_{\bullet}(Y).$$ 
Checking that this natural transformation respects sources and targets as defined in the OP is the only place where we use that $F \Delta^0$ is the terminal object.
The remaining question is why simplicial homotopies of simplicial abelian groups induce chain homotopies on the corresponding chain complexes. This is where the work in Hatcher's proof is, and it has absolutely nothing to do with $C$ or $F$. It exhibits part of the monoidal Dold-Kan correspondence, namely the part having to do with the Eilenberg-Zilber map. 
