I would assume that we are all here because we love dogs, so I'll ask this - if you took your dog to a veterinarian, would you prefer the vet and staff who were compassionate or those who were not? The vet who petted the dog, looked into it's eyes, took time to reassure you, while still doing all that needed to be done to care for the dog, or would you prefer someone who did all the necessary medical things but clearly did not care for the dog or for you?
As I mentioned above I managed a vet clinic for many years. One of the most telling characteristics of a good vet (and I've worked with many) is their ability to retain compassion despite years of seeing tragic things happen to animals. In clinics with more than one vet the busiest vets, the ones always asked for, were the compassionate ones.
I always managed to retain compassion and empathy for both the pets and the clients, and this resulted in many owners requesting that I be present for euthanasias of their beloved pets. In some cases I held the pets when the owners felt they were not emotionally strong enough to do it - and in some cases I held the owner's hands to give them support. And they didn't ask me because I was lacking in compassion, but rather because they knew that while I could not possibly care for their pet to the same degree they did, I did care. I shed some tears over these events, but I also managed to leave my work at work and not let it affect my home life.