Now, that last is just nonsense. We have animal shelters all over the place, and many vets and some pet stores function on a smaller scale as such as they can. TV stations in my area run "pet of the week" (usually dogs) stories to find homes for strays. All sorts of rescue organizations exist, many also run ads on TV. Over the past several years we (my immediate family) have taken in 3 stray dogs, and I've lost count of the cats -- the latest of which was near feral and would not come near people, but would eat my daughter's pet cat's food. One day the stray showed up with a bad head wound, and as he got weaker became a little more "accepting". We caught him, took him to a vet, and spent way too much money we could not really afford to patch him up, get him neutered, etc. Now, several months later, he's the biggest over-affectionate...
Point is, enormous effort is exerted in the US to find homes for strays. The trouble is, the problem is even bigger: Owners who think it's cruel to the animal to not let them reproduce, or, some just don't care and abandon them. At some point, the only humane thing left to do is euthanize some strays. Otherwise they starve, fall ill, get themselves shot raiding livestock (and I've had to do that, too), etc.