Why do the electrons, arising from the oxidation of the anode, flow through an external circuit? Why does it not just flow through the electrolyte and reduce the cathode? Why do the electrons give the wire preference?
The crucial point which explains this is that there are two different forms of conduction happening here.
The first kind of conduction, and the one we are most familiar with is electronic. This is a flow of electrons, and is the conduction which flows in the wire and through the load (a motor, LED etc).
The second kind is the flow which occurs in the electrolyte, and this conduction is ionic. It is a flow of charged particles (ions) within the electrolyte. Even though we often speak of the conductivity of an electrolyte, what is really meant is the electrolyte's ionic conductivity. This is governed by the mass of the ions, their concentration, temperature and the viscosity of the solution.
Electrolytes do not really conduct electrons at all, which is why there is no 'short circuit' through them. The conductivity arises due to several mechanisms - either the ions themselves move through the solution, or electrons (or protons) hop from one ion to the next, conveying the charge.
This will only happen if there is a means of causing the flow of charge through the electrolyte. When it is open circuit, the electrolyte is overall neutral in charge and no current will flow in it. If an external circuit is connected, it allows electrons to flow from one electrode to the other, and then the corresponding amount of charge will flow through the electrolyte to balance it out, producing current.
Electrical current will follow the path of least resistance (i.e. it will flow through whichever material is a better conductor); in this case, the external circuit. The conductivity of the electrolyte will depend on what's dissolved in it. I.e. pure water has very poor conductivity, while sea water (dissolved NaCl etc.) is a much better conductor. However, neither are as good as a metal such as copper, and hence electrons flow around the circuit.
If the electrolyte did happen to be a more effective conductor than the external circuit (can think of no examples), then presumably the current would flow through the electrolyte instead.
Hope that helps!