Here's a proposal: First, all dog breeding is regulated as strictly as bars with liquor licenses. Criminal charges and jail time if found doing it illegally.
Breeders and owners are required to have insurance coverage specific to dogs.
Owners are held liable for all damages their dog causes, no matter what. Your dog attacks somebody, mauls them and they seek settlement? It's on the owners. If the owner cannot be determined, the breeder is held liable.
Or, just make dogs above say, 30 pounds illegal. "But what if your dog is little and just really fat and becomes over 30 pounds?" Then you are fined and told to get your dog healthy within a certain timeframe, say 90 days, or surrender the dog and pay a fine for neglect, or get charged with animal abuse.
"Well then, nobody would want to get/breed dogs that are anywhere close to 30 pounds full grown because of the risk of violating the law." Yes...that's good. The only dogs people would have are small dogs that can't severely maul/kill people even if they do go off the rails.
"But what about hunters or sportspeople that use dogs to hunt or herd?" Plenty of herding/hunting dogs are within that 30 pound weight range, like Corgis or Beagles. Obviously there would be exceptions for government organizations that use large dogs for legitimate purposes like Saint Bernards for search and rescue.
He's a proposal: you don't need new Draconian laws every time something bad happens. That's how politicians pass every bad law. Patriot act ring any bells?