What is the logic behind the execution procedure in Fantastic Beasts And Where To Find Them?

I would wager that a death sentence in the wizarding world isn’t as quick and easy as one might suspect. Unforgiveable curses are exactly that- they are lawfully unforgiveable. Although we learn that some unforgiveable spells are mostly used for torture, we know that Avada Kedavra is ‘the killing curse’ and, although this would do the trick as it causes a painless, instant death, the effects of using this spell are strong for the caster. Not only that, but the use of any Unforgivable curse on a human would carry the punishment of a life sentence in Azkaban.

"The Cruciatus, Imperius, and Avada Kedavra Curses were first classified as Unforgivable in 1717, with the strictest penalties attached to their use." —- Professor Dumbledore's Notes from The Tales of Beedle the Bard

It is interesting to note too that, as Voldemort had utilised horcruxes and still died by Avada Kedavra, he described the event as ‘pain beyond pain’, and as if his ‘soul had been ripped from his body’, hence why the spell leaves only a dead body without any signs of physical trauma, much to the confusion of many muggles in history. This may be considered an unfit death, as it changes the state of the soul. Something the Ministry undoubtedly would choose to avoid.

As well as this, from the same source;

“This is not the only spell that can prove fatal; Fiendfyre, Sectumsempra, Confringo, Diffindo or even Stupefy in exceptional circumstances. Antonin Dolohov invented an unnamed curse that could also prove fatal. However, [Avada Kedavra] is the only known spell whose sole and primary application is death.”

I believe, as nothing is explained regarding the method of the death sentence, this is the most humane method to use, and one which has no accountability. There are no hangmen or guillotine operators, only a nurse who (very creepily) uses a spell to remove happy memories, placing them into the large basin (much like a pensieve, it seems), coaxing the person deemed punishable by death to sit upon the chair, which is then engulfed with a rising, egg-shaped form. It is supposed that this would kill her, however much like a cremation, this may simply be a respectable and tasteful method to ‘say goodbye’ to the person, to watch them happy one more time, before the person is removed from the room via the chair and egg, then executed in another location. At this, we can only postulate given the information, however it is safe to assume that this is a humane, painless death.

Source: http://harrypotter.wikia.com/wiki/Killing_Curse

Well, you'd think that the US/Muggle/non-Mage method of execution could be much simpler -

1. First an anesthetic is injected.
2. Then a paralytic agent is injected.
3. Then some form of drug that will stop the heart is injected.

You'd think that just whacking them on the head or throwing them into a tank infested with laser-beam equipped sharks would be simpler, but the complexity arose from general moral discomfort with the idea of execution, and the desire to somehow make it more "humane."

I'd guess the convoluted wizard/witch process arose from similar types of concerns and considerations, if you want an in-universe explanation of how that could happen.

The potion kills by encasing a person in a sphere then ingnting them. Perhpas unforgivable curses are not used or known in the US. Burning was once a way to execute witches. It could be a cross between electric chair and gas chamber.