Well, I don't understand the electrostatics, but here is another physical heuristic:

Impose a temperature distribution at the exterior, and measure (after some time has passed) the temperature in the interior. This gives a harmonic function extending the exterior temperature. [What's the electrostatic analogue? Formerly I had written "charge density", 
but now I am not sure if that's right.]

I think this strongly suggests a mathematically rigorous argument:  We are naturally led to model the time-dependence of 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.]