Of course it is their choice.
In the instant case, he knew he would be executed as you say and presumably accepted that as worthy martyrdom.
However, it was always the choice of the state to live stream it. And by doing so, they have reduced themselves to the level of Hamas, Al Queda and ISIS all of whom beheaded in the same way.
It is disgusting and beneath the dignity of a civilized modern democracy. I would hope that most Americans agree.
The most passionate people in discussing this are coming from two areas, far left liberals who are anti death penalty and current/former servicemembers who wish that they could be assigned to the duty of executing him. For one side it is the belief that every life is sacrosanct and for the other it is about settling debt to those who died, were wounded or had to suffer through what he caused.
Who is "civilized" is inherently subjective and heavily shaped by cultural, historical, and philosophical perspectives. Where you come from primarily determines what you believe is civilized. If we are doing the math of what people are the world actually think is civilized, you are going to be sad and disappointed because while you are in the Western majority, the majority of the world does not follow your beliefs. It is human nature to believe that we are part of a "civilized" culture and that everyone else is barbarians. I personally don't believe that I can really judge whether a person is actually a good person or not until I've seen them in high stress situations. Anyone can be moral when they don't have a stake in the game, but if you had to look in the eyes of Francheska Velez's parents and explain why you don't believe that the guy who murdered their pregnant daughter deserves death you may not be as exuberant as you are at your computer.
And while I was initially surprised at the decision to do it publicly, I absolute agree with it. We live in a world where you can get more prison time for saying mean things then some people get for rape. If there is going to be a death penalty, then due it in public instead of privately and let the world see it for all that it is. I'm not a fan of hiding the realities of life, I've raised my kids to understand that life can be brutal and that actions have consequences so they'll both be allowed to watch.
And I've already warned them, as I warn everyone now, that after the volley, an Army Dr will evaluate Nidal, and if he is still alive, the OIC/NCOIC of the detail will pull out their side arm and put a bullet in his head. The world isn't sunshine and rainbows and if removing him from the world gives any of his victims even the slightest bit of comfort, I'm all for it.