Inside the battery, electrons are indeed passed from one atom to another like jumping fleas.
But that’s not how electricity flows through a wire or through a complicated electric circuit. The electrons don’t just enter one end of a wire, hop from one atom to the next, and come out the other end.
Let’s say that the battery’s voltage is pushing electrons through a wire from left to right. What really happens is that each electron repels its right-hand neighbor, because they are both negatively charged and similar charges repel each other.
This nudges the neighbor toward the next right-hand neighbor, which nudges its neighbor, and so on.
By the time the wave of nudging gets to the other end of the wire, which is a lot faster than an electron can get there by broken-field running through the jungle of atoms, the effect is exactly the same as if those end-of-the-wire electrons were the original beginning-of-the-wire electrons.
Who can tell one electron from another, anyway? Not even another electron.