Indeed Don, I've also heard the 'drunken man' analogy - the electrons bumble along bumping into the material's atomic lattice.
The mean free path in something like copper is 10's of nm. If memory serves, the theory of the conduction mechanism is attributed to Drude (his "Free Electron Theory") and, at a top level, explains the mechanism of how conductivity varies with temperature (as the material cools the probability of collisions with the atomic lattice structure decreases, so the mean free path of the electrons increases). I say "top level" as Drude's work assumed a perfect atomic lattice- another chap (Matthews? or Matherson? something like that) refined the work to allow for impurities.
Also, as I recall, it is Drude's work that led to electron microscopy- if the electrons could "whizz" through atomic structure you'd not see much, analogous to 'over exposure'?
Having said that, the analogy of an electron flow is helpful in understanding electronics at the macro (circuit) level but I'm not sure I'd suggest anyone "pretends to be an electron", other than as an initial model for a total beginner.