Are we technically reducing electrodes/wires for the instant that an electron passes through them?
Reduction and oxidation are defined as gaining and donating electrons from the valence shell of an atom respectively.
Whereas in the case of electrons flowing in the electrodes and connecting wires the electrons are passing from the interatomic voids and not being released by the wires and the electrode.
(There will be some electrons which will interact with the conductor's atoms but what will happen is that they will get deflected. There will be a slim chance that a few electrons will replace an electron from the conductor's electron cloud but that doesn't change the situation as both the electrons are same in a classical sense. )
If your claim was true than the anode would have got some extra negative charge and the cathode extra positive which would have driven up the voltage higher than that expected from the cell alone.
In a similar way when we connect a wire it doesn't magically change the measured voltage.
Even when we perfom the same experiment with different inert electrodes and different kinds of wires the measured reading doesn't change much. (There are some changes in current flowing through the circuit due to the difference in resistances of different materials)