My textbook seems to suggest the free electrons in conduction band move left and reach the positive terminal of the battery. Does this mean electrons in valence band are not allowed to move left(hole moving right) and enter the positive terminal of the battery?

Similarly, it says the electrons from the negative terminal of the battery enter from the right directly into the valence band holes. Does this mean the electrons from the battery are not allowed to enter into the conduction band?

In summary: Electrons in an intrinsic semiconductor always leave from the conduction band and always enter from the valence band. This doesn't feel right. What am I missing?