Unlimited electrons?Can a conductor cancel a very strong field?
Compare the number of electrons in a Coulomb to the number of molecules in a mole.  There are a lot of electrons in something like copper.

Any E field that would require more than a tiny fractions of the electronics in a conductor to slosh from one end to the other, the E field would be so strong that it would arc and generally be unsustainable.

As a exercise, calculate the E field required to displace just one Coulomb from one side of a copper penny to the other.  You will see it would need to be unrealistically large.  Then compare that Coulomb of electrons to the total number of electrons even just in the outer conduction band of all the copper atoms in the penny.  You'll see that number is vastly higher then one Coulomb.