To your last point, I still think slaughtering with a <sharp> knife is more humane. You slit the animals throat, it's traumatizing, it's painful, but you've started the "bleed-out timer." The bodily processes know that's NOT GOOD, and plan accordingly. If it's anything like humans, perhaps the animal's body will dump large amounts of feel good chemicals near the end.

Juxtapose that against the airgun, a sudden impulse to the central processing unit of the animal. Chemicals and neuron activity get disrupted immediately, the animal may be comatose but not necessarily dead for some amount of time as the signal processing is just obliterated. Chaotic disruption of the system seems more unpleasant/inhumane than the more outwardly barbaric but ostensibly better slicing of the throat.