In a voltaic cell, taking one with $\ce{Zn}$ and $\ce{Cu }$ electrodes for an example, an EMF is formed because the equilibria at the $\ce{Zn}$ electrode is shifted more towards the $\ce{Zn^2+ + 2e-}$ side than the copper electrode’s equilibrium is shifted towards the $\ce{Cu^2+ + 2e-}$ side. Electrons, I understand, then flow from the $\ce{Zn}$ electrode to the $\ce{Cu}$ electrode and redox occurs.
The problem I have is that, if electrons migrate from the $\ce{Zn}$ electrode in this way, surely this will mess up the equilibrium at the $\ce{Zn}$ electrode and, even though it will attempt to shift back to counteract the change, the overall result will be a loss of delocalised electrons to participate in the $\ce{Zn}$ equilibrium. This would imply a decreased emf towards the copper electrode and hence an increasingly more positive $\pu{E_{cell}}$ value over time (assuming the $\ce{Zn}$ electrode to be on the left of the setup).
Am I correct or have I gone wrong and, if so, where is my mistake?