in theory this would work since it's a plotter. In practice every Etch A Sketch has its own minute mechanical flaws and idiosyncrasies which makes it so that the knobs sometimes lock up and skip at random parts which would make mass automation difficult. On top of that, each Etch A Sketch has varying line weights etc so it's hard to pick units that consistently function exactly the same both mechanically and cosmetically.

People have been able to program some really cool stuff with arduinos but this level of detail would probably take more effort than it's worth for reproducing.