Sweden did had a catch-and-kill program about a hundred years ago. It is mostly forgotten here, but it had two parts. A law was made that all dogs must be ear marked (today chipped) and listed in the national registry with their owner, and any dog that did not have a marking was killed. When a puppy is sold it must also have a veterinarian checkup, and the first thing any vet will do is to chip it and fill in the register data.
If a registered dog run away and get lost it will likely end up at the police, and the owner will get a bill for housing and handling. If you sell a dog, a breeder better make sure the registration papers get signed or the seller will be held liable. This way there are not much room for a feral dog population to get started.