Don’t forget what I think is the most important reason for elaborate “humane” execution methods. Those ordering or tasked with the executions don’t want to think of themselves as the “bad guys”. Historically what that means has varied strongly. But in our societies we have very little official appetite for being bad or cruel in a physical manner.

And then it’s a matter of required competency. It seems to require real skill to decapitate a person with a sword cleanly. Hanging cleanly is much less simple than it seems. Killing is fairly simple. Killing in a socially acceptable manner much less so.