We have animal shelters in Germany, and they do their best to give stray animals to families. They usually don't kill animals that are given into their custody, although I would say that they're a little bit faster with a deadly injection towards injured animals than a regular vet would be, just because they can't afford a 200 euro surgery on an injured animal, which someone who loves his pet would maybe willingly pay.
There are few stray dogs in Germany, mostly because German dog owners must pay a communal "dog tax", and dogs must be vaccinated and have a vaccine and "pet pass". But there are stray cats alot, and even the registration duty doesn't keep people from putting dog puppies simply out onto the street. But the shelters do a real good job. And they don't kill animals without a reason; they are even forbidden to do so by law.