While I feel this is a good question, I seem to have to deal with on almost a daily basis and have noted that the main issue with addressing it comes from a sociological aspect versus a biological aspect. The two sociological issues are defining the word “humane” and the cultural aspect of shooting a living animal while the biological issue is achieving death.
The problem with defining the word “humane” is that everyone is going to have a different answer as it will be based on that person’s moral beliefs first with the presented situation second. For some, their moral belief is that it is never appropriate to take a life of any living animal regardless of the situation. Hence, their definition of humane would be to let nature take its course and not interfere. For others, they may not believe in taking the life of an animal; however, if that is the only way to stop the animal from additional suffering, then they it may be acceptable provided it is done in a manner they approve off (i.e., chemical injection, shooting, etc.). For these individuals, their definition of humane is more along the lines of actions causing the least amount of suffering to the animal during the dispatch process. Then we move on to those that don’t believe in killing; however, if it needs to be done they don’t really care how it is done (i.e., drowning, gas chamber, blunt force trauma, etc.). Humane for these individuals is more about stopping any further suffering of the animal versus how it is achieved.
When you look at the majority of individuals, most do not have any issues with animals being dispatched provided it is done reasonably quickly with minimal pain and discomfort. Individuals that support hunting, fishing, trapping, and meat farming activities make up the majority of this group. This group’s definition of humane has more to do with how the animal will be treated during the dispatch process than with the actual taking of the animal’s life. The majority of individuals I know performing hunting, fishing, trapping and meat farming have the same belief as above regarding how the animal should be treated, but the difference is that they have to look at how the animal will be used before they can determine how the animal will be dispatched. Humane for these individuals has more to do with the choosing the appropriate dispatch method than performing the dispatch. Lastly, there are those individuals that have no issue with taking the life of an animal nor do they have any issues with how it is done. For these individuals, humane is more about entertainment and rights (i.e., human superiority) than anything to do with how the animal will be used or any suffering it must go through.
Now if we look at the sociological side of this it becomes very interesting as individuals can change the group and definition of human based on the animal the definition is being applied to. For instance, it may be wrong to dispatch any wild mammal with large eyes, but it is absolutely fine to kill livestock, fish, reptiles and insects. Likewise, it may be acceptable to kill wild animals, but unacceptable to kill cats, dogs or specific species. Additionally, how the individual performing the dispatch has been raised can account for what is considered acceptable and humane. These quick examples show how we have a large base of actions/beliefs that range from never to dead is dead no matter how long it takes or suffering involved based on multiple factors.
Specific to hunting and wildlife control actions, you need to start by looking at how animals die, how the dead animal will be used, and how it will be recovered. When focusing on shooting, animals die by severe disruption of the central nervous system (brain shot) or by loss of blood (everything else). Despite popular opinion, it doesn’t take 1,000 fpe to kill a white-tail deer with a bullet. Rather, a better way to look at this is it takes approximately 1,000 fpe for the bullet to perform the way it was designed to perform. If there isn’t enough energy, the bullet may not penetrate far enough into the animal and it may not expand properly whereas if there is too much energy you can get too much penetration and again it may not expand properly. Both of these situations can create problems, most notably lack of hydrostatic shock and a clean wound channel.
Hydrostatic shock is a shockwave that goes through the animal’s body causing trauma to tissues, organs, blood vessels and nerves around the bullet impact area. This can be important when choosing a bullet as the more trauma the animal receives, the faster the animal goes down. During times when there isn’t enough hydrostatic shock from the shot for whatever reason (bullet design, slower projectile from black powder or air rifle, etc.) you want a ragged wound channel as this increases the trauma resulting in more tissue being destroyed and thus more blood vessels being broken and torn making it harder for the body to repair itself. This is also why the military use FMJ (full metal jacket) bullets instead of hunting bullets (i.e., fragmenting). A FMJ bullet minimalizes tissue damage versus a hunting bullet that is meant to maximize tissue damage.
This concept is similar to that of using a broadhead on an arrow instead of a target point for hunting. If you shoot an animal with an arrow tipped with a target point, there will be relatively little trauma created as the wound channel will only be the size of the target point (think of stepping on a nail). On the other hand, if you use a broadhead you will cut a wound channel rather than punch one, and you will cause trauma away from the arrow shaft as well as getting better penetration (again, more trauma).
Taking this very quick explanation, we can now get back to the question. First off, most big game hunters will not take head shots since the head is the trophy and by doing so, they’d ruin it. Secondly, as already stated for bigger animals a bad head shot can result in unintentional damage to the animal allowing the animal to escape before any type of follow-up shot can be taken. As many hunters now have to pay (sometimes thousands of dollars) for the opportunity to hunt a location/species with the generally accepted rule (sometimes law) that a wounded animal counts as a dead animal, it doesn’t make sense to take a high risk shot. Third, when using a hunting bullet, the bullet is designed to mushroom or fall apart unlike an arrow or air rifle pellet/slug. This makes a chest shot with a hunting bullet the more ethical shot in many cases as an air rifle pellet/slug will perform similar to the target arrow analogy above. Lastly, big game hunters as a group are taught to not take head shots as it is such a small target and many bullets may not get good penetration before fragmenting. Instead, hunters are taught to wait for the classical broadside chest or quartering away shot instead where again, the bullet can perform as designed.
This isn’t to say that head shots aren’t ever taken as many hunters do so, but it is mainly with non-fragmenting or non-expanding bullets (this is the class most air rifle pellets fall in to) on small to medium sized game. For these animals (rabbits, squirrels, raccoon, etc.) head shots are encouraged as they are generally hunted for food or pelts rather than head trophies. In these cases, a head shot is generally preferable as 1) the smaller head size means that a miss does not carry with it the same injury chance it does with the larger head size of big game, 2) helps prevents the animal from not being recovered by dropping it immediately instead of having it run off and tuck into an area where it cannot be found, 3) eliminates damage to the meat so more of the animal can be consumed and 4) minimizes damage to the pelt resulting in a higher selling price.
My guess is that the group that most likely takes the biggest number of head shots on all wild animals are those doing wildlife control. These individuals have a need to drop the animal in their tracks that many hunters do not. Those doing this commercially or for a governmental agency may have to show proof they shot the animal as they may be getting paid/compensated by the animal and cannot afford to not recover chest/gut shot animals or waste time with extensively tracking them. Also, remember that most commercial operators and governmental personnel are being paid to remove multiple animals that are causing damage instead of using any shot animals for food or pelts. This can be different with individuals doing wildlife control on a noncommercial basis even though they also will be targeting problem animals, but as there is not the same economic reason for them to do so. Hence, they may be doing so for entertainment, skill improvement, or protection of property instead of offering a service or fulfilling a governmental task. Also, it is important to point out that many laws in the US do not allow for usage of the meat or pelt from problem wildlife taken by commercial operators and governmental agency personnel and thus the animal must be thrown away whereas many individuals are able to keep and utilize animals they shoot.