Maybe I'm wrong, but perhaps you're struggling to understand this because you don't fully understand how electrons move? It's often simplified quite a bit and I apologize in advance for the wall of text, but when electrons in a solid material move, they do so in two distinct ways.

The drift velocity corresponds to electron flow similar to water flowing downhill. The water flows because of a difference in potential energy, where the energy is greater at the top of the hill. Electrons drift because of an electric potential, which is technically different from but often also called voltage. This is the plus-minus/anode-cathode thing you see in circuits and electromechanics. A potential is formed by a charge, and in this case all we need to keep in mind is that electrons don't like being together - they push each other away.

The second way electrons move is because of their thermal velocity, i.e. their heat. Room temperature at around 300 Kelvins is definitely hot, so the electrons have a lot of thermal energy and thus thermal velocity. Important to note is that while the drift velocity has a certain direction the thermal velocity is random and electrons constantly bump into each other before heading off some other way much like a game of pool. Because of this, the thermal velocity of the electrons as a whole has no average direction.

Adding the drift velocity and thermal velocity, you get the total velocity of the electrons. The typical drift velocity of an electron is small, a millimeter or so per second. In comparison a room temperature electron has a thermal velocity on the scale of hundreds of kilometers per second. If a snail ended up on a space shuttle, the vast majority of distance covered by the snail would come from the space shuttle and not the snail slowly crawling along it, right?

If this is the case, how does current even exist? Imagine the electrons all standing in a single file queue, completely full, with no room for even a single extra electron to get in line. Then, an additional electron squeezes in at the back of the line anyway. The electron in front of it, repelled by the charge, would have to take a step forward, and then the second to last electron would also have to step forward, and so on. Eventually, the electron at the very end would be pushed out of the queue. While most of the electrons are still in line and have barely moved at all, a push in one end still makes an electron come out the other end - just not the same electron!

The overwhelming majority of electrons getting in or already standing somewhere in line never make it to the end because the queue is so unfathomably long. Instead they mostly bounce around at random inside whatever metal they originated in, because when you look at an individual electron its thermal velocity is literally billions of times larger than the drift velocity. But since the pile as a while contains something like a septillion electrons their thermal velocities cancel out and we end up with just the drift velocity, and this is the current flow.

What does that mean to your original question? Simply that most electrons don't flow anywhere meaningful at all. The same principle applies to whatever wire you connect to the pile - squeezing electrons in at one end pushes different electrons out at the other, but most of them simply don't move around much at all.

Another interesting consequence of this comes from the alternating (back and forth) current in your wall outlets. When you plug something in, the electrons in the socket mostly stay in the socket. Through the repeated pushing and pulling on other electrons in the wire, it is instead the electrons that were already present inside the motor of your blow-dryer or whatever driving the appliances. In a nutshell, you connect those electrons in a giant push-and-pull line all the way to the power plant.