I'm looking for an intuitive way to think about how electrons in a conduction band (or in an unfilled valence band) make a material a conductor, whereas filled bands result in an insulator.
What I think I understand:
A metal is a lattice of atoms with a 'sea' of delocalised electrons. If we consider these electrons as particles, an electric current can be thought of as the drift of these electrons in a particular direction when a potential difference is applied.
Energy bands arise due to the mixing of atoms' orbitals into molecular orbitals. The periodicity of the lattice results in band gaps forming. That is, certain electron energies aren't allowed.
The highest occupied electron energy band is the valence band. If this band is full (all $k$ states are full), the material is an insulator as an applied electric field has no effect on the states of the electrons (bottom row on figure) - there's nowhere for electrons to go as all states are taken. However, if the valence band is partially filled, only a small amount of energy is required to shift some electrons into higher energy states (top row on figure). Thus the material is a conductor.
What I'm confused about:
What's the link between electrons being able to populate new $k$ states, and the macroscopic property of conductivity, or a flowing current? After some reading of similar questions, I've come to the rough idea that if an electron can easily access empty $k$ states, it can "hop" around the crystal lattice relatively easily as it has empty spots to hop into. So different $k$ states correspond to different spatial locations? This is obviously treating the electron as a particle rather than a wave.
Is it more "right" in this situation to consider electrons as waves, thus filled bands result in standing waves in the electronic wavefunctions, and travelling waves for conductors? (I'm very unclear on this take of things.)