@darthtig,
Start with microchipping every single dog or cat that's born, so far as that is possible. This makes strays findable and helps to reduce the problem of designer animal theft (it happens). It also means that people would be accountable if their unaltered dogs or cats were out roaming the streets - which means the number of outdoor cats in particular would start to drop.
Yes, there will still be plenty of outdoor animals, and they'll have to be rounded up and, sadly, a lot of them will be euthanized. It's still a better ending than being hit by a car or becoming dog fight bait. I say this as a big-time animal lover, I might add.
Make spaying and neutering free. Make altered animals free from shelters, with pet food companies stepping up with serious coupons for food, treats, apparel (collars, leashes, and coats), toys, and beds. Quite simply, make it far more attractive to own an altered animal than an unaltered one. The only people who should have unaltered animals should be breeders and show animal owners and handlers. Yes, this means spaying and neutering every mutt in the land, unless they suddenly turn into the next fad, e. g. the labradoodle. And then labradoodles can be treated the same way as currently recognized breeds - altered unless being bred by a reputable breeder, or being shown.
In the meantime, also, work with breed rescues. Insist on laws that require breeder licensing and inspections, in order to shut down as many puppy mills as possible. Animal hoarders should go to jail. Backyard breeders should be fined. Like I said,
turn this economic. As it is right now, there are dogs roaming free on the streets of Houston because there's a culture there where people get tired of their puppy and send him or her out to face the world once the poor dog starts chewing (about 12 - 18 months for most dogs).The females end up pregnant while the males end up aggressive, and everyone gets a fair shot at being hit by a car.
All over, everywhere I have ever lived, I have seen outdoor cats. Make it harder to have an outdoor cat. Cats that live outdoors don't live as long as cats that live indoors. If cats were forced to live indoors, people would be a lot more likely to spay and neuter them, in order to avoid spraying, yowling, and all the other fun that comes from animals going into season.
All of these proposals would reduce pet populations significantly. And I would hope it would change some attitudes. We had a female pit bull for a while, a beautiful dog. Lots of people in our neighborhood suggested we breed her. We had her spayed. That should be the rule for most female pets, not the exception.