I'd really like to hear a complete answer to this question. I have my own idea and although I don't think it's a perfect answer to your question, perhaps it will help someone.

There's a way of defining cofibrant replacements in certain categories call the Boardman-Vogt $W$-construction. I know it from the work of Berger and Moerdijk on operads. I believe it was first defined topologically, but I only know it algebraically and for algebras it goes something like this:

The unit interval could be described by the ring $I=kt \oplus \, ks \oplus \, ke$, where $e$ is concentrated in degree $1$, and $d(e)=t-s$. $t$ is the identity and $s$ is idempotent, also $se=e$. There's a unique augmentation. So this is a chain complex modelling the unit interval. The multiplication could be described as taking the maximal value, this is how it's defined topologically. $t$ corresponds to the maximal element $1$, $s$ to the element $0$.

Now let $A$ be an associative algebra, the $W$-construction $W(A)$ looks very much like the free product $A*I$ (perhaps it's the same I don't remember), with the induced differential. $W(A)$ is a cofibrant replacement for $A$.

This contains a subalgebra quasi-isomorphic to $W(A)$, which may be given by the bar-cobar construction. Remember that the bar complex $BA$ is given by taking the free coalgebra on $A[1]$, with a square-zero coderivative induced by the multiplication map. The cobar complex $\Omega \,C$ on C is the free algebra on $C[-1]$. This carries the square-zero derivative defined by the coalgebra structure.

That's a mouthful, but all you need to remember is that the bar and cobar functors define an adjunction, actually a Quillen equivalence of model categories. And the unit and counit of this look very much like the $W$-construction.

In Berger and Moerdijk they do this not for associative algebras but for operads. And there are similar constructions for any operad (generalising the case for the associative operad above). In one example the bar complex is based on the free colie algebra and the cobar complex is based on the free commutative algebra.

Onto the categories of modules: there is an adjunction between the category of modules of $A$ and the category of comodules for $BA$. This is actually a Quillen equivalence.

Now back to the Koszul property. An algebra is Koszul when the bar complex is quasi-isomorphic as coalgebras to something very small, in many cases quasi-isomorphic to its homology. Call this nice coalgebra $C$.

Then there is an adjunction between the category of modules for $A$ and the category of comodules for $C$.

So to me at least the Koszul property concerns situations when large resolutions of things, say $\Omega \, BA$, may be replaced by much smaller things. And this means a lot for the categories of (co)modules for the objects concerned. But that's just my personal impression.