It's odd that you would say that the humane society doesn't euthanize animals. Maybe not all of them do. In Orlando, where I used to live, they told me that they euthanize kittens if they don't find a home within 3 weeks, and adult cats have 2 weeks. I don't know their policy on dogs.
When I moved here, I found a stray dog. I was worried that it may be owned by someone and be lost. I thought that if I lost a dog, the first place I'd look for it is at the humane society. I called them and asked about it. The humane society in my town is run by the local vet and his wife. She said that people rarely ever call them for their lost dogs. I asked what would happen if the owner doesn't call. She said they'd try to find it a home. I asked what would happen if they couldn't find a home for it. She said they put them down after about a month.
If there are human societies that don't euthanize their animals, then I have never encountered one. In fact, a lot of nonprofit organizations are out there that "rescue" animals from humane societies before they're put to sleep. I used to be involved in an organization that spays and neuters ferrel cats. The reason we didn't just take them to the humane society is because in large cities the odds are that the cats will be euthanized. I would check on it before just taking an animal to one.
I don't know anything about horses at all. I never pretended to. I just know that several people around my house own horses, and people are always firing guns. Those people must be really inconsiderate neighbors. Then look at the old west movies where people fire guns around horses like it's nothing. All I know about horses is what I've seen in movies.
I have a question on this matter, though. If a dog is running loose on my property, but not really harming anything, can I shoot at it to keep it off of my property? I only ask because of one particular dog that chases a little cat that uses an area under my house as a hideout. The cat isn't mine, so I can't say it's harming MY property. Most of the dogs around here don't bother her.