Protons are tightly bound to the nucleus of atoms, and the nucleus is 'shielded' from the outside world by the 'cloud' of electrons around it. It takes a lot of energy to separate a proton from the nucleus. When that happens, it's a 'radioactive' or a 'nuclear' event, and the atom has changed into an atom of a different substance. This isn't something that's happening around us very often.
On the other hand, the electrons are on the outside of the atom, and it's rather easy to convince an electron to leave its atom and flow off to somewhere else.
Electrons flow as charge carriers in metal wires because they are loosely bound to their atomic nuclei, making them easy to move. In contrast, protons are tightly held in the nucleus and cannot move freely, thus they do not participate in the conduction of electricity. The free movement of conduction electrons allows them to create an electric current when a voltage is applied.
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