iirc almost all "hunters/predators" have the instinct to kill as quickly as possible, not out of ethics but because killing slowly risks injury, the prey escaping, or something else coming along and nicking your dinner.
Yes true, but I was referring to humans.
I'm not quite sure what you're getting at here. Why would you intentionally wound an animal with a gun of all things, just to slit its throat? It sounds like whomever you went deer stalking with was a rather terrible marksman. You do NOT intentionally shoot things in the leg just so you can slit their throats. A deer (or pretty much anything else) should be taken down with a head shot, or else a heart/lung shot. Then and only then would you bleed it out and dress it.
, head shots are a must! with the appropriate caliber to do the job.