I'll try to answer your question, sorry if it doesn't satisfy you

Executions of officials or military commanders isn't unusual. Other than to root out failed, unruly, incompetent, or otherwise troublemaking officials, it also served to paint the Ottoman Sultan and its Dynasty's image as a just and ruthless state that always expected the most out of his subjects and that none of its subjects were beyond the Sultan's attention. Those who had earned the Sultan's satisfaction or had provided a high standard of service for him will be rewarded with honours, favours, and rewards in a public or semi-public ceremony; and accordingly those who had failed him or earned a disfavour were subject to dismissals, confiscations, banishments and executions. In this case, a failed or disastrous military campaign was especially severe because it will result in a loss of face. However, not all who had failed gets executed. Another important thing to play in whether or not someone gets executed is their support in the court - those who still had patrons or obtained favours amongst the courtiers in the Porte could summarily escape execution. One example of this is Grand Vizier Halil Pasha, who even though returned to Istanbul defeated by the Safavids managed to keep both his head and rank by managing to secure a favourable term with the Safavids, therefore salvaging his reputation in court.

Now with that out of the way, let's examine Kara Mustafa's execution, because it will correlate to what have i explained above. When the deputy of the Grand Vizier reached the Sultan's camp in Edirne bringing the news of defeat and further plans for a counterattack, the Sultan was furious and the deputy was met with an angry rant:

Both you and your chief (the grand vizier) have joined forces among the band of the accursed. He has destroyed my good name and reduced my honourable reputation to ruin. He has crushed my good soldiers, killed my good commanders and allowed the infidel to take hold of my lands

As you can see, the defeat pretty much was seen as a loss of face and had costed Kara Mustafa good will from everyone in the court. After executing the deputy for bringing bad news, the Sultan summarily ordered his execution. Imperial secretary Ahmed Agha's diary described his execution, which i will take the translation from Marc David Baer's Honoured by the Glory of Islam:

The grand vizier is depicted as being resigned to and eager to meet his fate. On Christmas Day, he was stripped of his imperial seal, the banner of Muhammad, and keys to the ka‘aba which he hung around his neck, all signs of his being the representative of the sultan and leader of jihad. When he was told of his impending end while engaged in afternoon prayers, he declared that what God wills happens. He finished his prayers and told his pages to leave but not to forget him in their prayers. He took off his turban and ordered the executioner to enter. To be considered a martyr, he had the rug removed so his body would fall in the dust. The executioner raised the dismissed grand vizier’s beard and passed the noose around his neck. The grand vizier admonished him to tie it well. The executioner tugged at the ropes two or three times, and the former grand vizier expired. The body was taken out to an old tent, where they prepared the corpse for burial, performed the funeral prayers, and then cut off the head, buried the body in the courtyard of the mosque opposite the palace, and sent the head to Mehmed IV in Edirne. First it was placed at the palace gate to serve as a warning to others, then it was buried in the courtyard of a local mosque.