Current is the result of a difference in electric potential between two points. If an entire object is at the same potential, then there's no reason for current to flow anywhere over, under, around, or through it. That's how an electric-repair lineman goes up in an insulated bucket, connects himself and the bucket to the line, and then proceeds to work on the live line without danger.
The difference in electrical potential across the entire bird is the difference between his feet ... perhaps 1 to 10 inches along the length of the line that's hundreds of yards or hundreds of miles long. The potential difference between those two points a few inches apart is very small.
A bird sitting on a wire is safe because there is no electric potential difference between its feet, preventing current from flowing through its body. However, if it touches two wires simultaneously, it creates a path for current to flow due to the voltage difference, leading to electrocution.
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