I don't really get why you, amongst others, would find such nature-imitating way of killing an animal "inhumane".
When killed by a predator in the wilderness, animals usually suffer a much more brutal death, compared to a quick throat slitting. As some member of this forum already explained, animals such as lions take considerable amount of time to finish their prey.
I know I should avoid talking about things I've never experienced, but I have this feeling that death can only truly be embraced and accepted by the body, mind and soul either at a "terminal" age, or in great suffering (terminal disease, physical assault, inescapable fate such as falling from a cliff,...). To put it better: at a moment in an individual's journey when living is ultimately compromised.
That's why you sometimes hear stories about people who shot themselves in the head and are still wandering the earth in the living's world, as spirits. Because they died physically, but their soul -or mind, or whatever- did not have time to realize and accept the fact that they are now dead. And so the person/spirit still thinks it is alive, although it is not in reality (in that case it is advised to make contact with a medium so it can guide the wandering spirit to "the light").
I feel that such death as described more above is necessary to truly depart from the living world, and join the other side.
Also it is known that a great amount of a molecule called DMT is released at the moment of the individual's death (I think it is also released in great quantity at birth, and a bit while in the dreaming phase. But I don't know a lot about it.). I suspect that if the person dies too instantly (eg: bullet to the head), DMT doesn't have time to be released, and the person might not have the time to "enter" death, embrace it, if you believe that both material compounds such as DMT and immaterial states such as death interacts.
However I am not saying that any living creature should experience unnecessarily long time of suffering before death, even if I do believe that the body is intelligent enough to know the moment when producing the experience of pain to fight or flight is just not appropriate to the situation anymore (last option: freeze). While, again, being aware that some deaths probably hurt a lot more than others (being set on fire in a cage, as an eg taken from recent events...)