Ordinary homology and cohomology factor through chain complexes via singular homology and cohomology. What about other (co)homology theories?

That is, for each spectrum $E$, do we have a lift in the following diagram?

$\begin{array}[ccc] & \mathsf{HoTop} & \overset{E}{\to} & \mathsf{GrAb} \\ & \underset{?}{\searrow} & \uparrow \\ & & \mathcal{D}(\mathsf{Ch}_{\mathbb{Z}}) \end{array}$

Where $\mathsf{HoTop}$ is the homotopy category, $E$ is $E$-homology or $E$-cohomology (in which case it's contravariant, of course), $\mathsf{GrAb}$ is graded abelian groups, $\mathcal{D}(\mathsf{Ch}_{\mathbb{Z}})$ is the derived category of chain complexes in abelian groups, the functor $\mathcal{D}(\mathsf{Ch}_{\mathbb{Z}}) \to \mathsf{GrAb}$ is homology, and the functor labeled "?" is the desired lifting.

It would just about suffice to do this in the universal example of stable homotopy ("just about" only because we have to do this for all spectra now), because we have a factorization:

$\begin{array}[ccc] & \mathsf{HoTop} & \overset{E}{\to} & \mathsf{GrAb} \\ \downarrow & \underset{\pi^S}{\nearrow} \\ \mathsf{HoSp} \end{array}$

where $\mathsf{HoSp}$ is the stable homotopy category, $\pi^S$ takes stable homotopy groups, and $\mathsf{HoTop} \to \mathsf{HoSp}$ is either $E\wedge(\Sigma^\infty-)$ (for homology) or $\operatorname{Fun}(\Sigma^\infty -,E)$ (for cohomology).

So in some sense I'm asking about the relationship between two sorts of graded-abelian-group-valued decategorification: taking homotopy groups of a spectrum, and taking homology groups of a chain complex.