My understanding of Matthiessen's rule is that when electron conduction in a solid is inhibited by more than one mechanism, that the overall resistivity $\rho$ is formed as if these resistance source were added "in series," $\rho = \sum_i \rho_i $, where $i$ enumerates the distinct resistive phenomena. This is easy for me to conceptualize by associating each resistivity with a scattering rate $\rho_i\propto\nu_i$. It seems reasonable then that the total resistivity is proportional to a total scattering rate $\rho\propto\nu$ formed from a weighted sum of the individual collision rates.
Now let's say I want to abstractly represent a solid as a circuit element, where each scattering mechanism is a resistor that contributes to the overall resistance of the solid: one for scattering off of ions, one for scattering off phonons, etc. The argument above says I should lay these resistors in series.
If we didn't have that knowledge, though, one might hypothesize that for a homogeneous solid, the electron current is split up equally amongst the different scattering mechanisms. This would lead to a circuit representation with the different scattering mechanisms' resistances added in parallel. Is there a way to deduce that this is wrong without appealing to kinetic theory ($\rho_i\propto\nu_i$)?