I guess I see it differently than many: euthanasia is one of the most valuable tools we have in veterinary medicine. We can peacefully, humanely end the life of an animal, often ending or preventing suffering. I don't see what greater gift you can give a patient with terminal illness or unmanageable suffering. (FWIW...cattle sent for food are not euthanized by veterinarians but by trained plant personnel, although vets are present on the floor.)
@GUH, I find it interesting that you perceive the role of a physician to only be healing. You mention your oath, so I imagine you think, "First, do no harm" is the leading principle... but isn't allowing a person to suffer doing them harm? And what about the harm you do by prescribing certain drugs (such as chemotherapeutic agents) or performing invasive surgeries, for example? Attempting to heal a patient is always the goal, I certainly agree. But when it becomes clear that the person is suffering and will continue to do so for any length of time, the option to end that suffering is a form of healing, too, both for that person and for their families left behind.