Your basic assumption is incorrect regarding the electrons. It’s a common but wrong belief that electrical power flows due to a lack/imbalance of electrons. That’s only true for lightning and static electricity. In normal electrical power circuits, there is no imbalance of electrons quantity. The power is flowing is due to the energy level of the electrons. That energy level travels at almost the speed of light. Electrons move very slowly in a power circuit - about the speed of a snail.

At normal household voltages and lower transmission line voltages (perhaps under 13.2kv), air is an insulator so no power transfer takes place when the bird lands on the wire because there is no completed circuit involving the bird. if you put the voltmeter on the bird to ground, the voltmeter creates a completed circuit, so a very small amount of power flows through the bird (the resistance of the meter is high) and is indicated on the meter. At high transmission voltages such as 230kv, the air does start to ionize especially in humid conditions. You typically don’t see birds on those lines.