Other mistake: In the "Uncut" version of the film, after the "Dinner Party" at Valentine's place, when Valentine is at the door saying goodbye to Harry, he says "Just give me a couple of days to think over your proposal." His lisp has disappeared on the word "days." (00:57:00)
Other mistake: During the pub fight scene, the guy who got hit with the glass cup (Wound on his forehead), pulls a six shooter out of his pants, but he fires eight bullets at Harry's umbrella.
Other mistake: When the Kingsmen are looking on-screen at the dossier of the princess, the attached photo of her is just a screenshot from the earlier scene where she is at the dining table with Valentine and the Swedish prime minister, just before she is imprisoned. Her photo was not taken at that point, and couldn't have been attached to her private file.
Other mistake: In the final scenes where Eggy goes to have a fight at the pub with his mother's boyfriend they are listening to Slave to Love by Bryan Ferry. The counter says 962 whilst the tape machine is playing (electronic), but the counter does not change.
Answer: It was to test whether the candidates were willing to do whatever it takes to get the job done. If they were willing to shoot the dog they had been caring for, it means they would be willing to make a tough choice in the field to complete a mission.