That's just not how electricity works. I have analyzed this stuff myself with very precise equipments. I'm not just drinking the kool-aid here. When the switch is turned off, no noticeable current is flowing through that wire. Nothing that a consumer grade EMF meter would even dream of being able to pick up anyway.

Think of it like water. Imagine you have a pipe with water flowing constantly from one end to another. If you put a junction in there with another pile, the water would now split and flow through the two outlet pipes. But now put a cap on one of the outlets. Now the water inside that pipe is completely still, even though water is still flowing to the other outlet at the original pace.

This is why fire sprinkler systems have to be regularly drained and the water in them is so incredibly disgusting.

Electricity works the same way. It's flowing through the wires in your walls, but when the switch is off on the lamp, it is not flowing through the wires on the lamp.