The way i see it is, if all dogs are spay or neutered, and No one is breeding. Yes that would help with the overpopulation. But would that not cause the canine species to become extinct, since there would be no purebred or otherwise to breed?
If all ethical breeders(breeding to preserve the line) stopped breeding then only byb puppies or puppymill puppies would exist. Would this not damage the bloodlines and lead to medical or mental issues in the breeds?