I don't think bird don't get electrocuted because they have high resistance...
I mean we are still talking about 20kV or so. If they did somehow complete the circuit, current would pass through them.
But here is my view.
They are NOT in fact completing the circuit.
If a bird sits on a wire, there is no way that it can complete the circuit, it is just sitting on a bare wire. That wire might be hot from the passing current, but anyway this is the same as you jumping and holding the phase wire, or touching a live wire, with just one hand.
But this only goes for low power lines, like 20 kV. For 500 kV, there is a different story.
And no they do not act like a resistor. Do an experiment at home.
Take a 9 V battery, connect it to a resistor, let's say 500 ohms, and then short circuit that resistor, in parallel.
Current won't go through the resistor any more, as a poster before said, current is lazy and it wants to go through a path with less resistance, ergo your short circuit wire will get very hot.
Your bird is your resistor, and your line is your short circuit.