Question

Why do electrons rather than protons make up the flow of charge in a metal wire?

Answer 1

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.