Far more children are hurt by their parents every single day than those hurt by dogs. Do you suggest we bring in exams and licenses before people are allowed to become parents?
Utter piffle (although there have been calls for natural parenthood - for want of a better phrase - to have the same checks as adoption)
Licences, microchipping and 3rd party insurance should be the minimum requirement with a small reduction on the licence fee for having pet health insurance and / or attending dog ownership and training courses. These could be offered by the kennel club, breed specific associations and the RSPCA.
Policing of this? The licence fee could fund something like the old dog patrols, higher fines for not picking up after your dog would also add funding to dog wardens (who could check microchips & insurance the eay the police can check insurance & mot on cars)
And yes, as with guns and so on, the criminals would ignore it, but the vast majority of owners would comply. Societal pressure in not accepting rowdy dogs etc would also play a part. If it's going to go on your record more people would comply.
Then tackle unlicensed puppy farms etc. Rehoming a fog should come with transfer paperwork as with selling a house or car. It's not impossible.
Added powers for the RSPCA not to just tackle the low-hanging fruit would help.
Responsible dog owners shouldn't find any of this too onerous, surely?