In one respect, you don't have that much power. Much of the time, all you're doing is making suggestions to owners about what they can do to improve their animal's health or condition. You can suggest an operation or x-ray, but they have to approve it. When they don't take your professional advice, you wonder why they came to you in the first place, but ultimately it's up to them.
On the other hand, you are saving lives. That is kind of the ultimate power. Maybe you shouldn't be playing God, but it's way too much fun to put on the robe and beard and speak in that deep, booming voice. All dogs go to heaven, anyway.