I guess a lot of it depends on the persons emotions. Some of us are more inclined to end suffering since we can't stand the thought or sight of it, some us are more of the let nature take its course crowd. To each their own is how I see it. As mentioned we all hunt for different reasons, my main reason is meat plain and simple, horns are a bonus.
As for me if I had a tag, the elk would have never been chewed on by the coyote, because I would have been quartering it up by that time.
Without a tag, I would just snap a few pictures and report it to fish and game.
With my luck though, the elk would have walked up on me still smoldering and twitching
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Boarding a spaceship would also be pretty cool to.