The beheading is awful, and given Saudi Arabia's track record with human rights abuses, I'm more than a little skeptical of the possible processes through which this capital punishment was arrived at.
That said, I think three issues are being mixed up in this thread:
1) Whether capital punishment itself is ethical, acceptable, or humane
2) Whether beheading is an appropriate way to carry out capital punishment
3) Whether the death sentence imposed on the woman in the video was arrived at through a "proper" trial and legal proceedings.
Now, the first question is very complicated, and the answer can depend on a lot of variables, in my opinion. I think it's important to remember that Saudi Arabia is far from the only country in the world to have capital punishment. Unless one criticizes all other countries that have capital punishment, I don't think it makes much sense to criticize Saudi Arabia as "barbaric" only for having capital punishment. If capital punishment is unethical, inhumane, or unacceptable in any other way, then that applies to all countries, not just Saudi Arabia.
The second question is also not going to have a uniform answer, obviously, since many people don't object to beheading as a method of capital punishment while others do. I personally think the most important question to ask about beheading as far as its being a method of capital punishment goes is how painful or slow it is. This is hinged on the premise that the answer to the first question of whether capital punishment is acceptable at all is a yes (which I'm personally not sure of one way or the other, although I lean toward no in general).
In the U.S., capital punishment is mostly done using lethal injection, and we can see stuff like this:
In the USA, a number of lethal injection executions have been botched. Some executions have lasted between 20 minutes to over an hour and prisoners have been seen gasping for air, grimacing and convulsing during executions. Autopsies have shown severe, foot long chemical burns to the skin and needles have been found in soft tissue.
Source.
Let's put aside our initial reactions to the graphic nature of the video for a moment, although that may sound bad at first. If we do that temporarily and consider the above questions, how is beheading much different from most other methods of execution? Would being shot hurt less? Would being hanged? I don't think what matters here is what the execution
looks like; I think what matters is what it
feels like to the person on the receiving end of the punishment. If being burned alive inflicted no pain whatsoever on them, no matter how gruesome it looked, I think it would be a more ethical method of capital punishment than the lethal injection (again, this is hinged on the premise that capital punishment is acceptable at all, for the sake of discussion).
This also ties into the "in broad daylight" aspect of the execution: I definitely view public execution as inhumane, unethical, and uncivilized, but one of the objections I saw brought up to the video was something along the lines of, "While the police are watching." Yeah, that's capital punishment; it's state-sanctioned killing. Name one country where capital punishment is used where law enforcement doesn't watch as the convicted are executed. How is police watching a beheading any worse or better than a law enforcement person hanging or shooting a convicted person without much hesitation? Is this possibly also about the fact that beheading perhaps
looks more gruesome than hanging or shooting?
The third question, whether the punishment of the woman in the video was arrived at through a full-blown trial and legal proceedings, is one that I think isn't easy to answer, but I'm personally quite skeptical of the justice of the Saudi legal system. Let's not forget that as far as capital punishment goes, this is the same system that deems homosexual sex, apostasy, and "blasphemy" capital crimes. I find it unreliable and overwhelmingly irrational at best.
My last thought in this post is that I think the fact that the source of the video is Saudi Arabia will probably make some people's objections to it even stronger, be it due to general distrust of Saudi Arabia or opposition to Islam, or any other number of factors that are tied to the country, its culture, and/or its religion rather than the acceptability or lack thereof of capital punishment or the method used.
Add the above factors to the fact that many people objecting to the video already oppose capital punishment regardless of the method or country doing it and you have a pretty strong recipe for outrage—which I'm not saying is or isn't understandable per se; I'm merely saying that if we don't consider the source or graphic nature of the video for a moment, what we end up with isn't very different from the usual questions many people ask about capital punishment in general, not just when it happens in Saudi Arabia. It's not like many people would find it exactly pleasant to watch a video of someone being executed by hanging or the firing squad either, after all.
