You can argue against any law by saying "the law-abiding will follow it but the people causing the problem won't bother".
You could, if you wanted, set up a (probably slightly seeming, compared to what we have now) system where all dogs have to be chipped and owners have to be trained, licensed and insured, and it's actually properly monitored and enforced, with info on a centralised database on things like:
- registered owner and contact details
- other nominated persons with responsibility for dog
- registered home address(es)
- registered veterinary practice
- DNA info
- history (health, parentage, previous owners)
- veterinary statement of conformity to law on permitted breeds/types, certification of acceptable temperament, explanation of any tail/ear/other modifications, spay/neuter status (and vet explanation if intact), certification of suitability/unsuitability for breeding if relevant (with heavy penalties for breeding from animals not certified as suitable), vaccination status, etc.
- up-to-date dog licence fees paid
- third party liability insurance details
- up-to-date training and competence certificates for owner (unless they're only the official owner, and won't be alone with the dog without another nominated person) and all other nominated persons
- anything else useful
and the lowest possible threshold required to allow dogs to be scanned for a chip and check info, with scanners provided to any and all officials or designated individuals who might plausibly come into contact with a dog, and full legal right for them to do so, so it's considered no more of an accusation or an imposition than having your numberplate scanned by police or a car park. Neighbour complains to the council about barking, your dog could be scanned; someone says your dog looked at them funny, your dog could be scanned; someone anonymously reports your dog's ears look too short, you have a social services/HV visit, you let your dog off the lead somewhere questionable, a police officer walks past and fancies scanning it, dog could be scanned. Ideally some kind of improved chip/scanner so it's not a bulky extra thing to carry.
And although people on MN always pop up to say it's crackers, DNA testing of dog shit and dog bites has been done before now and could be implemented. Billed to the owners or their insurance when identified, or paid from dog licence fee revenue if not.
Any dog found without a chip, or which is not under the control of an individual who is able to prove license-holder/nominee status for that dog (or under the control of a qualified individual like a registered kennel, licensed dog-walker or vet), or with any info incorrect, absent, or not up to date, the dog is seized, and owner pays a fine, and either rectifies the problem or the dog is humanely destroyed.
Dog causes a problem, and depending on the problem, fines are levied, additional training is required to keep the licence (at the owner's cost), liability insurance kicks in, owners and/or relevant nominated persons lose their right to hold a dog licence or be nominated persons either temporarily (and must obtain a new training and competence certificates) or permanently, existing criminal penalties apply, dog is removed and rehomed, or dog is destroyed.
I know it sounds harsh to talk so much about destroying animals but I would like a much lower threshold in general for destroying pet animals, because I care about them and don't believe in gratuitous suffering. I don't think that the never-put-a-healthy-dog-down stuff does pet animals a lot of favours at all. So many of our pets are deliberately bred from unsuitable parents to be in constant discomfort and pain, or they're put in shelters where they're warehoused while they slowly develop psychological problems, or they're passed from owner to owner. They don't really understand the future, or dread the concept of death. They were brought into being to be pets, so if they're not being pets and they're in pain or stuck in a shelter anything beyond the short term for administrative purposes, it's pointless to inflict continued life on them based on the principle that maybe one day they might become a happy pet, IMO.
I would also want to have schemes in place to assist those who would genuinely struggle with the cost or the admin through no fault of their own, like homeless people, those with mental health problems, or people with unexpected changes in circumstance, so I'd set the licence fee nice and high for the majority to subsidise that help.
In the context of the overall cost and admin of having a family dog, this kind of system shouldn't be too onerous for most dog owners, and there's not much over and above what responsible owners already do. Owning an animal for pleasure that can, if not cared for and managed properly, result in injury and death to others, fear and intimidation of members of the public, fouling of the local environment, serious noise nuisance, traffic hazards, significant suffering of the animal itself, and so on, is a privilege and a responsibility more on a par with running a car than with anything else I can think of right now. These kinds of regulations and intrusions and fees would still be far less than owning a car.
And yes, there are still some people who run cars without insurance or MOT tests, or with bald tyres, or who drive recklessly and maim and kill people, so there would still be arseholes breeding massive powerful dogs, cutting bits off them to make them look 'ard, fighting them, using them as status symbols and threats and weapons, who would try and circumvent the system, but we could at least try to deal with it.