solid state physics - What does the concept of "free electron" mean in the context of band theory?

Often when conductivity is explained through band theory, the term "free" tends to crop up. As an example, I often come across descriptions of the valence band as a highest filled set of states occupied by electrons bound to their specific atoms; raising them to the conduction band supposedly "frees" them so that they can move freely in the metal, thereby enabling them to contribute to a current when an electric field is applied. In fact, in my introductory solid-state physics book, the additional electron contributed by a donor in a doped semiconductor is referred to as "loosely bound" to the donor ion, requiring a push into the conduction band to break free and become a charge carrier.

At the same time, I was also given to understand that the electrons of a valence band do not contribute to a current when there is an electric field, because their respective velocities balance each other out perfectly; there is no net velocity and hence no net movement. Raising an electron into the conduction band essentially means creating a hole in the valence band so that the electrons can now redistribute (in k-space) and thereby achieve a non-zero net velocity.

But according to this latter statement, the electrons in the valence band should contribute to a current across the metal when an electric field is applied.

A) How then can the valence band electrons be bound to a specific atom, as the former statement claims, if they are simultaneously capable of acting as charge carriers? Also, how then can the donor electron - which occupies an energy state above the valence band - be "loosely bound" to the donor atom, when the electrons below aren't?

B) Assume that we raise the temperature enough (without the metal somehow disintegrating) so that some electrons from even the lowest band leave for higher energy bands. Will the holes that are left behind in this lowest band also mean that the remaining electrons in this band can carry charge, similar to how the electrons in the valence band with holes were able to carry charge?

I'll be grateful for anything that can help me clear up this mess!