From the Crown Prince Faction's perspective, I could see them wanting to hype up the trial to overthrow the Duke. They get House Kendall's support, and now have a good shot at eliminating the Second Prince's faction.
They needed the trial and the declaration of an execution to get nobles on their side; probably with the argument of "See what kind of stupid bullshit the Second Prince and the Duke will pull if they get into power? It might be YOU and your family next on the chopping block with an idiot like that on the throne". That's why the trial and sentencing was necessary, but not the execution itself, and why she could be spirited away when the execution was to be carried out.
As for the King, I could see him letting it play out if he wanted the Duke removed, and to see which of his two sons has the political cunning and charisma needed to grab the throne from their brother. After all, it's a Duke; not even a King can easily get rid of a Duke. He would have needed a lot of shady dealings and wetwork done to weaken a Ducal House, which would have led to destabilization of the country and might have given the Duke enough time to make some countermoves. Having this kind of "apparently spontaneous" conflict happen amongst his vassals, caused by the Duke in question overstepping what polite society considers to be his bounds, let's him keep his hands clean.
Of course, this is a lot more thought than likely actually went into creating this one-shot, but I think it can stand up to some scrutiny.