Here is a sketch. First, here is the argument for untwisted homology that I want to base the argument for twisted homology off of. Every categorical thing I say below is $\infty$-categorical by default, e.g. every colimit is an $\infty$-colimit and so forth.

The untwisted $R$-homology spectrum of a space $X$ with coefficients in a ring spectrum $R$ is the colimit of the constant diagram $X \to \text{Mod}(R)$ with constant value $R$. Taking $R$-homology spectra defines a functor from spaces to $R$-module spectra which is itself cocontinuous (because colimits commute with colimits), and in particular which sends pushout squares to pushout squares. But $\text{Mod}(R)$, being stable, has the property that pushout squares are also pullback squares. A pullback square of $R$-module spectra can be converted into a fiber sequence of $R$-module spectra, and then we can apply the long exact sequence in homotopy.

The argument works essentially without modification for twisted $R$-homology, except that the domain category is no longer spaces but, say, pairs of a space $X$ and a local system of $R$-module spectra on $X$ (there is no particular reason to restrict our attention to local systems of $R$-lines). Again taking $R$-homology is a cocontinuous functor and hence again sends pushout squares to pushout squares, which again are also pullback squares.