The key to the answer is where you equate a lack of electrons with an abundance of protons. That is a very misleading analogy. The correct analogy is actually to equate electrons (carriers of a negative charge) with holes (which is the absence of an electron where one should be. Holes are positively charged).
Protons are fixed in place (at least in a solid, and if you ignore Brownian motion and the like). They are "frozen" into the nucleus of each atom.
One gram of copper, or one centimeter of copper wire, contains a specific number of atoms, and therefore a specific number of protons, and that doesn't change no matter what.
Now in a neutral substance, there is one electron for each proton on average. Individual atoms can lose one electron, or even two, and that makes them positive ions. However, when that happens, these ions have a strong attractive force on electrons. Removing the first electron from a neutral atom tends to be easy (in a conductor). Removing a second one becomes more difficult, and removing a third electron, or even more, becomes incrementally more difficult, and eventually impossible simply because the atom becomes more and more positively charged, and therefore attractive to electrons.
But hypothetically assume that you could remove all electrons from your conductor, and you could prevent electrons from the surrounding air to get back in. The first thing that would happen is that all the remaining atomic nuclei would repel each other. Your conductor would disintegrate (probably with a huge explosion).
Now if you could prevent that, too, you still couldn't have an electric field. An electric field exists between two charges. An electron does not "have" an electric field. A proton does not have an electric field.
So if you could remove all electrons from a conductor (which you can't), and could prevent it from flying apart (which you can't), there would be nothing left that could generate an electric field.