Are these just totally different electrons?
Yes. The electrons actually don't drift far. Typically only a few millimeters a second, if I remember correctly. Compare it with a traffic queue; when the first one moves, he leaves space for the next, who then moves and leaves space for the next and so on. The motion progresses through the whole queue (wire), even though the individual cars (electrons) don't move much.
New electrons enter at the other end, since they see a lack of electrons here, now that they all have moved a step forward. A lack of electrons at a spot is namely a spot with slightly less negative charge, which repels electrons less than all other places. So new electrons will quickly move here.
If so, then what's the point of the battery?
We just described how electrons move. But why they move is another question. Something must "pull" in the electrons, otherwise they wouldn't want to move at all.
Since we know that negative charge is attracted by positive, we can create such "pull" by placing a positive charge. That's the battery's positive end. Electrons are drawn towards it, and that makes the drift. That causes the current.
As electrons keep arriving at the positive end, they would pile up. They would accumulate a bigger and bigger negative electric field, which would repel and soon prevent any further electrons from arriving. The current would stop again. So the battery has to carry them away.
Inside the battery some complex chemistry takes the incoming charges and carries them to the negative end. They don't want to be here, because they are repelled from something negative. So they drift away into the ground, making space for others.
A battery is like a pump in a water pipe that keeps the flow going.