You need to separate the concept of voltage from current.
The canonical example is a hose connected to a faucet. For simplicity, assume that the hose is already full of water, but the faucet is off, and the hose does not leak (i.e. it is aimed toward the sky).
Now assume that you just barely turn on the faucet. A given molecule of water (representing an electron for the analogy) will take quite a long time to propagate from the faucet to the end of the hose. This is like "current".
However, the water molecules at the end of the hose will almost immediately begin to fall out of the hose. Importantly, these water molecules are not the ones coming from the faucet, which are still way at the other end of the hose. You could say that an "aqua-motive force" from the faucet is pushing the water out the end of the hose. This force travels much faster than a given water molecule. This is like "voltage" (or "electro-motive force")
To recap: current represents discrete "somethings" moving past some point, and voltage represents the force which is pushing those somethings. They move at vastly different speeds; I believe that while voltage propagates through copper at roughly 2e8 meters per second (approximating for permittivity), the actual electrons move at a glacial 1 meter per hour (and that's assuming DC, AC will move even more slowly as the electrons "slosh" back and forth)
Why is this? Well, in a vacuum (or a superconductor) the electrons would fly by at about the same speed that the voltage propagates. But in real world conductors, the electrons slam into other electrons, into a nucleus, they get excited by thermal energy, etc. So they actually end up bouncing back and forth at high speeds, individually, but collectively they all move slowly in one direction.
The voltage never has this problem, it's just an electric field. That is why we use voltage and not current for signaling. The signal from across the Atlantic (ignoring fiber optics) is not carried by individual electrons, but instead is sensed as the force which pushes those electrons. That is why electronics is the art of directing voltage, not current. The wire is never "empty", it is always full of electrons; the question is whether there is a force applied or not.