I think this issue is really hard to find a middle ground on.
On one hand we have the massive overpopulation problem, and as someone that works in a shelter (and plans to base my career in shelters), it is extremely difficult to look at the results day after day. When you are personally faced with the problem, and are physically seeing animals living in cages for months and being put to sleep when they go kennel crazy, any solution is going to start to look good because the status quo is so terrible.
And I think this desperation for a solution is something that is hard to understand for people that don't work in shelters, because it's easy to forget that it's happening, and even reading that X million animals are euthanized every year in shelter does not have half the weight that being present for one of those euthanasias does.
On the other hand, I think anything backed by AR groups is dangerous and is leading toward an undesirable goal. In addition, the results of this kind of law are going to vary a lot by location; where I live there are many people that do not have the means to leave the city via car and bring back an animal, but many of the animals here are from BYBs and oops litters anyway. So the effect is going to depend so much on the size of the city, the mobility of the residents, what the surrounding area is like (I'm near the puppy mill capital of the northeast - Lancaster PA - other areas may not be so saturated with puppy mills), what laws are already in place, and so on. I don't think that those of us living all around the country can really judge what the effect would be in San Francisco, and be able to state whether or not it's easy to leave the city and so on, because we don't live there.
Anyway, the argument on this thread has mostly stated that this law is a bad idea. That's fine, but where we are right now is pretty awful itself. The Philadelphia city shelter gets 30,000 animals a year, and half of those are euthanized. That's one city in one state. So yes, it would be bad to have no pets, but I think it's just as bad that we're killing 15,000 pets a year in my city alone, and it's mostly fueled by puppy mills, pet stores, and backyard breeders. So if we don't like this solution, it's not enough to just say it's a bad idea. We need a better idea, and we need it yesterday.
Whippets is right that we need better enforcement of existing laws, and I'm glad he or she brought it up. Part of that is financial - getting better funding for humane law enforcement is a step. My shelter does humane enforcement for the entire state, and gets zero money from the city or state. That should change. Law enforcement is not enough though - there are so many "oh she got out and came back pregnant" litters, and those aren't illegal or cruel, they're just pointless and sad.
Spay/neuter resources are huge. So many people at my clinic cannot afford to have their animals fixed. "Well they shouldn't have animals" you say, and that's a nice thought but they do have animals, and their animals are having babies all the time. Why not have the money put into enforcing these new laws be put into spay/neuter programs? Why can't cities fix animals for free for people with certain incomes, rather than inevitably house the offspring of those unaltered animals? Education in this area is so important too. People still think that having one litter is healthier, and have no concept of how expensive it is to vaccinate and deworm a litter of puppies every three weeks, or of the consequences of creating yet another litter of mutts. This is where veterinarians should be talking talking talking to their clients. Even if you're recommending against spay/neuter (that's already been beaten to death on these forums) tell people how to tell if an animal is in heat, and how to keep them separate, and warn that an animal may try to get out to breed, etc.
Legally, we should be thinking about laws that would solve the overpopulation crisis, while allowing for the purposeful creation of healthy, friendly pets that are bred toward ideal health and temperament. We need to look at ways to stop puppy mills and backyard breeders while allowing reputable breeders (and I mean breeders that health test, know the breed, provide humane conditions, screen homes, and compete or work their breeding stock) to continue to breed high quality animals. Maybe do a modified form of MSN for mixed breed or unregistered animals, or make it much harder to get a kennel license (and illegal to sell animals without one), or make it much more expensive to license an unaltered animal, or place a tax on pet sales that feeds into a low-income s/n fund. I'm just thinking off the top of my head and there are probably issues with those ideas, but I think it's counterproductive to just shoot down ideas without putting any more up. And I feel very strongly that anyone going into the veterinary profession should spend some time in a shelter and see where the shelter people are coming from and see the homeless animals face to face, and see the conditions that the puppy mill animals come out of, and realize that this isn't something that should be ignored. It is no coincidence that a lot of the arguments on this forum end up between the shelter people and the non shelter people, and I think a lot of arguments would be resolved by seeing why the shelter people get so up in arms about overpopulation and its causes.