Sorry, now I think I understand. I think the Wikipedia article is wrong (or rather, misleading).
The harmonic function to be "physically" produced is not the Coulomb potential, but the charge density inside a conducting medium. Since in equilibrium there can be no electric field, the charge density is harmonic.
So, impose a charge distribution on the exterior, and measure (after some time has passed) the charge density in the interior; that gives the solution to the Dirichlet problem. This is probably roughly equivalent to: impose a temperature distribution at the exterior, and measure (after some time has passed) the temperature in the interior.
I think this strongly suggests a mathematically rigorous argument: We are naturally led to model the time-dependence of charge density (or temperature) in the interior. This satisfies a diffusion (or heat) equation, but in words:
"After a time \delta, the new temperature is obtained by averaging the old temperature along a circle of radius \sqrt{\delta}."
This process converges under reasonable conditions, as time goes to infinity, to the solution of the Dirichlet problem. Anyway, we are led to the Brownian-motion proof of the existence, which I personally find rather satisfying. Another personal comment: I think one should always take "physical heuristics" rather seriously.
[In response to Q.Y.'s comments below, which were responses to previous confused remarks that I made: neither the electric field nor the Columb potential is a multiple of the charge density on the boundary: the former is a vector, and in either case imagine the charge on the boundary to be concentrated in a sub-region; neither the electric field nor the potential will be constant outside that sub-region.]