Decapitation and gallows with a proper drop height were both designed to be fair, in the sense that everybody killed would get the standard execution experience. It was meant to be an improvement to the status quo of plebian criminals getting tortured to death while aristocrats were given swift deaths.
Both systems are humane compared to the earlier status quo, but an even more humane system could certainly be devised. To answer your question, in the case of decapitation there is reason to believe death is not instant but rather that the severed head may remain aware for a few seconds. This is faster than hanging, but still not instantaneous.
A shotgun slug straight to the skull would be pretty damn close to instantaneous, but that's got a few problems. In addition to the psychological damage inflicted on any witnesses to the execution, it will have more inconsistent results. What happens if the shot is poorly aimed and 'merely' removes somebody's jaw bone? Done correctly, a shotgun slug would provide a more instantaneous death. However it's a less foolproof system. That's just an example though, it's easy to conceive of systems that are simultaneously more instantaneous than a guillotine but are simultaneously are more reliable.
Edit: re inert gas: Yes, though again there is the matter of reliability. An inert atmosphere will kill a human reliably, but the apparatus used may fail to provide an adequately inert atmosphere. This probably wouldn't happen with a mechanism designed specifically for execution, but it does happen a lot in cases of attempted suicide. Inert gas suicides will sometimes be botched, with the victim receiving just enough fresh air to survive the experience (typically with severe brain damage caused by the oxygen deprivation.)
Ultimately I think the whole matter is about risk tolerance. How much risk of executing an innocent man do you tolerate? How much risk of a botched execution do you tolerate? How much risk of releasing a guilty man do you tolerate? These are questions with subjective answers, and for many people their answer will be to tolerate no risk, e.g. abolish the death penalty. Other people are willing to tolerate more risk, so for them the answer might be "bring back execution but modernize the methods". Generally people on one side view the thought processes of people on the other side as alien or obviously defective, but I think that's generally not the case.
I don't understand why they don't anesthetize them before their execution. Put them under as if they are going in for surgery and then issues of inhumane treatment go away, and opens the way for far easier and less gruesome types of execution.
Turns out that proper anesthetization is more difficult than most people realize, especially since the competent medical professionals won't help for ethical and moral reasons. Also, drug companies won't sell the drugs if they are going to be used in an execution.
Since none of us knows the experience of death, it's not knowable what effect anesthesia has. Perhaps being numb to the experience makes it worse and not better.
The destruction of brain tissue during the experience of death, while bringing about a swift end to detectable life, may not be a pleasant way to experience death.
1) How much damage a brain can take before it's incapable of experiencing.
2) How fast a brain can have an experience.
3) How long it would take for a supersonic shotgun slug to reach point 1.
I'm extremely skeptical that point 2 is fast enough to meet the deadline imposed by point 3. I suppose I can't say it's impossible, but it seems very unlikely.
For that matter though, let's suppose point 2 is fast enough to meet the deadline. We don't actually know if that is a more unpleasant death experience than being hanged or decapitated. We can speculate about which experience is the worst, but none of the dead can actually report back to us with real data. My personal suspicion is that decapitation is the most unpleasant, since there is a strong possibility you will be conscious long enough to experience your situation, and that experience will be truly unpleasant. But who knows.
I don't think it's reasonable to make the assumptions you make in your three-part calculus, though I do acknowledge that it's a thoughtful way of considering the matter.
But what if an intact brain is an important part of navigating the first 10 minutes or so of the experience of being dead?
I prefer not to have my brain mushed at all, and let it very slowly decompose. Certainly the first hours of being dead, while it's still possible that electrical or chemical activity in the brain is ongoing, I prefer for it to be intact and at peace.
My calculus is predicated on the assumption of a materialistic mind and no afterlife (e.g. there is no experience of being dead, only experiences of dying.) If those assumptions are wrong, then I think there is no firm ground on which to speculate at all (maybe Valhalla is the true afterlife and a gory death in battle is the most form of death the dead find most satisfactory. I don't believe that, but I disbelieve it only as hard as I disbelieve the Christian afterlife.)
In my model of the world, perhaps a brain slowly ramping down is a pleasant experience. That's totally conceivable to me. However the brain being instantly disassembled faster than it can have experiences would be the absence of experience, be it a pleasant or unpleasant.
Gallows are higher impact than a guillotine (the body is accelerated by gravity, then rapidly decelerated by the rope.) So I don't think it's totally inconceivable that gallows might render somebody unconscious faster than a guillotine. However in practice I don't believe it actually plays out that way.
