I've appreciated reading (and re-reading) all 7 pages of this as it currently stands and agree with much of it.
The word I'm not seeing a lot of is "responsibility". People have approached it with comparisons between stalking and slaughterhouses and the conditions / approaches there, but I think the reason it pleases me to shoot and kill an otherwise - one assumes - happy and contented animal is "responsibility".
By that I mean literally, the responsibility of saying "I killed this to eat it" as well as the moral responsibility for the animal's death.
It's hard to make this argument without falling into a religious viewpoint - something I don't actually possess myself - but it does illustrate the approach all the same. Imagine one discovered at some point, be it at our own deaths or otherwise, that there is actually a creator God to whom we have to account for our actions. I'd far rather stand in front of him and say "I chose to kill this creature because I wanted to eat it and use it for my own benefit; I selected it, looked it in the eye and killed it, whilst ensuring it didn't suffer needlessly" than say "I let the slaughter man kill for me" or perhaps worse "I killed it because it was an inconvenience, not because I had need of it".
I think that when you get down to it, most humans are killers in one way or another, and more than that we're the most efficient and capable predator the world has ever seen. Once you get past that realisation, there is only what we kill, how we kill and the reasons we do so. I actually think that at the heart of most "antis" as we like to call them, is the inability to accept that killing is a fundamental part of human nature - it's what we're made / what we evolved for (depending on one's viewpoint).
I suspect a large part of the loathing these "antis" apparently hold for us is really a loathing of themselves - they know deep down that our success and our very identity as a species has come about because we, above any other species, were cleverer, faster and all round better killers than any of the species they might try to protect. It is only the acceptance of that fact - without prejudice - that can release any of us - hunters or antis - from any guilt we might feel either way on the issue.
To return to the how, what and why questions, compare the hunter who shoots a deer and wastes none of it, with the nonchalant swatter of spiders and flies. It is hard to argue that any spider or fly - even the unpleasantly poisonous ones - are a genuine threat to humans when they are so easily trapped and removed or killed. Why do some of us run around in fear of them, hysterically batting them with rolled up magazines at every opportunity? Are their lives worth less than ours, or less than a deer's? A hard question, but my instinct says no, though I haven't the time or effort to argue that point.
The deer hunter and the fly-swatter are both killers. The moral judgement is therefore not about whether to kill or not but how and why. A deer killed instantly with a bullet and a fly squashed with a magazine are probably equally humane killings, though there are other methods available for both tasks which would be less so. We can therefore judge the morality of the method of killing according to the degree of suffering caused. It is reasonable to argue that it is more moral to cause less suffering - most humans will concur with that (although no moral system can be absolute, except by consent). A slaughterhouse may be less humane; other methods even less so. Perhaps this is why the killing of geese with Gin, precipitating an agonizing death from liver failure, is specifically prohibited by law?
The "why" of the killing of a deer and the killing of a fly can also be judged from a moral standpoint. If I kill the fly to use it's carcass to feed a carnivorous plant, or to further laboratory research or to achieve some such similar aim, I give the killing a purpose. The same is true if simple convenience is achieved - I do not have a fly bothering me by buzzing round my head because I have killed it. However, that purpose may be judged less worthy by an observer, if it seems to fail to value the life taken. The how and the why are interconnected, of course, but it remains to us to justify not the killing itself but the means and the intentions by which it was achieved.
I applaud the OP for trying to get to the bottom of this and to construct something useful to us to explain why we do what we do. Ultimately, I agree with those that say there is no "proof" which shows why deer stalking is a moral good. I don't think an activity, per se, can carry that kind of moral argument. However, I do believe that the way in which an activity is carried out and the true intentions of those carrying it out are areas in which a moral judgement can be made: our strength, if we have any upon which to stand is that the vast majority of us stalk deer for the best reasons (ranging from food production to the outright spiritual) and try always to do it in the best possible way (humane, clean killing). Most of us, at least, do not kill deer (or anything else) simply because it is convenient or without reason, or approach the killing without giving the animal the respect it deserves.
On a personal note, it is the internalization of this approach which allows me to rescue bugs from pavements or rooms in which they have become trapped, where they might otherwise be killed (by others) to no good purpose; or which allows me to find fascinating deer, geese and the many other species whose appearance and behaviours interest me, and yet still pick up a gun and kill them in their turn. My work colleagues cannot, for instance, understand why I refuse to let them kill wasps, and then risk picking them up to remove them from a room: I ask them what purpose the killing would serve when it can be removed and released without harm and they never have an answer. They do however respect me as a hunter, and - I hope - as a moral person.