The delocalized electrons in a metal move about because of the electrostatic attractions between them and many different positive ion cores. But why does the same not occur with the ion cores themselves? Why are electrons, which bear the same magnitude of charge as nuclei, mobile, while the nuclei themselves are not?
The crystal structure of the metal holds the atom in place. BTW, since electrons are shared in a "sea" of electrons or conductivity band, no one atom is necessarily ionized. The crystal structure is also important for nuclear physics, e.g. using Mössbauer spectroscopy to measure gravitational effects.
Individual electrons don't actually move around the whole circuit, even in DC applications. Instead, think of it as a large stack of electrons; as one is pushed on the stack at one end, electromagnetic forces pop one off the other end. Though electrons do move a bit, the drift velocity is much less than the speed of light (in fact, it is somewhat slower than the last-placed snail in a race).
Compare the mass of a single electron and the mass of an atom, or the mass of an ion in a plasma. Note that acceleration of a particle is inversely dependent on its mass. This is important in fusion research, where confining electrons and ions is a bit tricky.
See this PPT presentation on metallic conduction.