I don't agree that stopping all breeding is needed, but I certainly can sympathize with the sentiment involved!
I think that dog ownership should require people to get their dogs neutered or take out a kennel/breeder license. If someone wants to keep unneutered dogs, fine, let them pay for it. The money can go to shelters and rescues. I think if a person has an unneutered dog, they should pay a premium for the privilege of keeping it.
I think that in urban areas, multiple dog ownership should be highly regulated. This would help to end dog fighting. Part of the reason dog fighting exists is because the dogs are inexpensive. If they were expensive, people would turn to other forms of gambling, hopefully those that do not involve live animals.
If someone wants to keep multiple dogs, they can pay for it. I don't think targeting a breed works, because fighters will just switch to another breed or say their dog isn't that breed, invent some other registry if need be.
In Holland, last time I checked, a dog license cost over five hundred dollars a year(I think it's much more now), and anyone who had a dog, had to take the dog to an obedience class and the dog had to pass an obedience test to get the license. Rural, urban, everywhere. I think that's a great idea, but wll never fly here, as people would object for myriad reasons, mostly their freedoms being trampled.