I just want to add that there is a reference for something related to that in Section 6.3 of volume 1 of Borceux's Handbook of Categorical Algebra. There is certainly some work to be done in order to turn it into an answer to this question and there may be some significant differences between the two cases, but this is as close as I can get at the moment.

In that case you should probably ask for the categories of elements of $D(G_i, F -)$ to be sifted for some chosen generating family $(G_i)$.

@local The simplicial set I associated with $(X, A)$ is not actually a simplicial abelian group and it does not detect local homology isomorphisms (as far as I can tell). You would have to suitably close it up under linear combinations, but then the resulting functor is no longer a right adjoint and it's not clear how to use it to create a model structure. I still think that existence of such structure is plausible, but probably it would have to be constructed using methods that Bousfield originally used to construct non-local homology localizations.

I did add an answer, but deleted it later since I realized it was incorrect and didn't know how to fix it.

@DenisNardin You are right. I got a little confused by these pushforwards at first, but they are just left Kan extensions along the induced functors between fundamental groupoids. In that case the construction of the model structure seems rather easy. I will add an answer.

Is this category cocomplete? It's fairly easy to construct filtered colimits, but I don't see why pushouts exist.

I don't understand why $\mathcal{C}'$ inherits a symmetric monoidal structure, the associativity and symmetry isomorphisms are barely ever identities. (And while associativity can be strictified, symmetry typically cannot.)

If we take the discrete topology on $\mathbb{Z}$, then topological $\mathbb{Z}$-modules are exactly topological abelian groups so that seems like the most natural choice. This way my construction describes free topological abelian groups on topological spaces.