I have been, since I was right out of babyhood, very close to all manner of critters. It is an integral part of my nature. Stray cats and dogs come right up to me. Horses, even cattle, have come right up to me in the field. I have tamed and placed many the feral-born kitten. Even my pet lizards know me from other people and run up to the front of their tank when I approach. I can "read" an animal, from a horse to a fish, and I basically know what is on their mind. That is why I was so successful at breeding even difficult tropical fish, amphibia, lizards, etc. That is why I haven't been bitten by dogs or cats who gave other people problems.
To me, the trust of an animal is something precious, it is a gift. To use their trust to enable me to approach them to kill them is something I could not do; the very practice fills me with horror.
Your high level of empathy explains much. A great many hunters become “tuned” to their prey, and have empathy for it. That likely seems weird to you, but hunting is a complex activity, one that can be engaged in with considerable nuance.
Hunters know that everything is mortal and that everything dies. Death by a human hunter is usually much cleaner and quicker than death in the wild.
When I am hunting, I feel completely alive, a part of the natural world. In many cases it is a transcendant experience; the death of prey is only a minor part of the process. Hunters often have a pang of regret. Many societies celebrate this dichotomy with rituals that give respect to the prey. Hunters know that life cannot exist without death.