Let $X$ be a nice finite cellular complex (a regular CW complex or a simplicial one), equipped with a local system $\mathcal{F}$ of free rank 1 modules over some Noetherian commutative ring $R$. What might be the most natural way to define the chain complex associated with this object?

In the majority of textbooks I saw authors follow the original recipe by Steenrod, and start by choosing arbitrarily a reference point (or a leading vertex in the case of simplicial complexes) in each cell, and then twisting the boundary operator by the module authomorphism corresponding to the homotopy class of paths connecting the respective reference points (and contained within the closure of the cell considered).

I have an impression, that at least for finite cellular complexes it is possible to define the chain complex with local coefficients in a more natural way. Let us think of $\mathcal{F}$ as of a locally constant sheaf of rank 1 free modules over $R$. In the stratification of $X$ associated with the cellular structure, let $S_d$ stand for the stratum consisting of the interiors of $d$-dimensional cells. Topologically, $S_d$ is a disjoint union of a finite number of $d$-disks. Let us define $d$-chains as global sections of the inverse image of $\mathcal{F}$ with respect to the inclusion $S_d \hookrightarrow X$ (the resulting module is naturally the product of copies of $R$, but since there are only finitely many of them, we can think of it as of the direct sum). Now, for any couple of incident cells (a $d$-cell and one of its $(d-1)$-faces), there is a natural isomorphism between the modules of global sections of pullbacks of $\mathcal{F}$ to them. To get the twisted differential it suffices to use this isomorphism (instead of explicit basises in $C_d$) in the definition of the boundary operator.

Even if this construction is limited to finite cellular complexes only, this way of getting rid of arbitrariness still seem to me to be too easy. Is there any flaw in this reasoning?

PS: The condition of finiteness of the cellular complex is indeed necessary, as can be seen from an example of a real line considered as a cellular complex consisting of 1-D cells $]n, n-1[$ and 0-D cells $\{n\}$ for $n \in \mathbb{Z}$. The pullback of $\mathcal{F}$ to the union of 1-D cells is a product of countly many copies of $R$, and thus contains a non-zero constant global section, which is a non-zero cycle in the considered construction. Maybe this can be fixed by treating the local system $\mathcal{F}$ as a cosheaf?