In npn transistor, why doesn't some base current flow in conduction band?
That's the whole point of P-type doping &mdash; the dopant atoms "sop up" any free (conduction band) electrons in the base region, because the overall system energy is lower if their outer shells are filled.

Ignoring the injected electrons for the moment, the only way for charge to move through a P-type semiconductor is for valence electrons to jump from one atom to the next, which looks like "holes" (the absence of an electron) flowing in the opposite direction.



Response to edit:

Theoretically, they do have three choices. But the relative probabilities of the three choices make the third option essentially impossible.

The diagram may be misleading. In a real transistor, the distance across the base region (between the emitter and collector) is many orders of magnitude smaller than the distance between there and the external base connection.
The probability of recombination rises exponentially with distance.