Most county and city animal control organizations do have a euthanasia option. Although there are a lot of private non-profit programs that will work with them to help get the animals removed and placed with fosters so euthanasia can be avoided. There are also just some animals that the animal control officers pick up that are just too sick to live a quality life and those will also get euthanized.
If you want to work with animals and don't want to see them put down, you might consider being a pet sitter. I did an internship at a vet clinic and we had to euthanize animals all the time. Even as a zookeeper I've been exposed to animals getting euthanized although it's much less frequent as a zookeeper.
I agree that it's sad when euthanasia is used as a last resort for an animal that can't be re-homed. However I've personally witnessed pets get so sick with cancerous inoperable tumors that euthanasia is the most humane to do.
Courtney recommends the following next steps:
- research online the various occupations working with pets (pet sitting; zookeeper)