I'm not convinced that breed-specific dangerous dog legislation is the best route to go — I agree with others who propose tighter legislation and regulation around ownership. We need a situation where owning a dog that may inconvenience, worry or harm other people or their property is a risky and unappealing prospect.
Compared to vehicle ownership, things like injuries, deaths and degradation of the environment are on a different order of magnitude, but IMO similar principles apply. You want to own something which can cause inconvenience, distress, financial costs, injury and death to other people. Therefore you should have to prove you're competent, pay through licensing for a system of regulation and enforcement, carry insurance in case of harm to others, and face legal or criminal consequences for any negative effects on others.
Ideally I'd go for something extremely strict, to make ownership of potentially dangerous dogs less appealing.
For example, you could require licensing of both owners and keepers, and separate licensing of breeders, with anyone who will have sole responsibility for a dog at any point needing to complete courses and pass tests, and keep their certification up to date. Anyone who's not old enough to follow the course and pass it isn't old enough to walk/look after the dog. Owners would have extra responsibilities and requirements, breeders yet more.
And separate licensing/registration of the animals themselves, with compulsory microchipping expanded and unified to provide a single registration number and contain extra information like DNA fingerprinting, and confirmation the dog has passed regular behavioural, temperament and training tests, plus a compulsory collar in public carrying the dog's registration number with a minimum font size.
A culture of routine checking of collars and microchips, with police officers, social workers, vets etc. carrying scanners. DNA testing of dog shit and dog-inflicted wounds, and retesting of dogs if any suspicion of discrepancy. Dog wardens patrolling. Large fines and criminal prosecutions for causing inconvenience to others by e.g. fouling, incessant barking, people having cause to fear your dog, jumping up on people in public, having the dog off lead outside designated areas, taking your dog somewhere it's not allowed, damaging someone's property or spoiling their food, clothes etc. Confiscation of animal plus loss of owner's licence for refusal to allow scanning, failure to chip or collar the animal or to keep it up to date, discrepancy between chip and collar, etc.
Dog destruction for any injury to people, pets, protected wildlife or livestock, or for dangerous failure of the behavioural, temperament and training test. Owner's insurance to automatically reimburse NHS for all human treatment costs, or private care if the victim prefers, plus compensation for lost income, suffering, mental health treatment and so on. Insurance to pay for all veterinary care for animal victims, financial and emotional impact for the owner of the animal victim(s). Insurance to pay for all property damage, including spooked or injured livestock. And any injury caused by a dog, owner/keeper prosecuted as though they caused the injury themselves. Your dog kills someone? Decades in prison for you. And so on.
(For the dogs' welfare, I'd also put limits on dog rescues, long quarantines and long kennel stays, with a recognition that euthanasia does not hurt the animal and it's preferable to keeping an animal warehoused. And no more imported street/rescue dogs — we have enough as it is. Plus, I'd have prosecution for anyone deliberately breeding or buying dogs with breed-related, inborn health problems, including all the flat-faced ones that can't breathe properly. I'd give existing animals an exemption (if sterilised), but all animals born after the cut-off and considered to have a high likelihood of such issues would be euthanised. If you only punish the breeding, there's still incentive to illicitly breed them, but if people know there's no point buying a mutated, suffering pedigree dog because you can't register it, get a collar to show it's licensed, or take it to a vet without it being destroyed, and then you'll be prosecuted for indirect animal cruelty on top, then I doubt they'd still be so popular.)
Not all of this stuff would work, some of it might seem draconian, and some of it would be expensive or difficult or both, but I do think we need a massive overhaul in how we think about the privilege of dog ownership, and how we manage it to reduce the impact on everyone else. Essentially, for all the many, many impacts dog ownership has or can have on other people — random members of the public, neighbours, everyone, and all so the dog owners alone can enjoy the benefits — I'd be trying to put at least the monetary costs of those things back on to dog owners, paid for by the dog licence fees and any fines levied. And as part of that, make owning a potentially dangerous dog much more difficult, because of the behavioural/temperament tests, and also something that could cost you all your money and your liberty, as well as your small children (since that alone doesn't seem to be enough).
I know that there would be people who would try to/manage to circumvent it all, just as there are people who drive without insurance or without a licence, or drive in illegal ways. But where something is a personal choice that can cause significant inconvenience or harm to others, we can at least try to prevent or offset the harm, and put the costs of doing so on to the people who want the privilege.
I also know that there are a lot of decent dog owners out there, with well-trained animals which have no detrimental impact on other people, who would object to paying for licence fees, training fees, testing fees and compulsory insurance premiums to cover all of this, but at the moment it's all of society that pays the cost of recreational dog ownership, rather than just dog owners.