Dog licences in the past have just been a fee which people paid, or didn't, much like a TV licence. Paying a sum of money does nothing to make people more responsible.
A dog licence run like a driving licence (education and a test) would improve matters but is hardly a panacea (look how many people drivers kill per year...)
If we're being honest though, it's not dogs in general, XL Bullies and American Bulldogs are vastly overrepresented in the figures - out of 16 fatal dog attacks in the UK 2020-22, 9 have included those breeds or crosses thereof
en.m.wikipedia.org/wiki/List_of_fatal_dog_attacks_in_the_United_Kingdom
Those breeds have a nasty tendency to attract the sort of person who wants to intimidate others with their dog's presence. Betty, 86, who wants a companion, does not acquire an XL Bully, she gets a bichon frise.
Breed specific legislation has been really poorly implemented over the last 30 years, and has led to very young puppies and adult dogs alike being put down without having put a paw wrong. Compulsory sterilisation and a temperament test to determine if they should wear a muzzle in public... I could get behind that.
Dogs being off lead has fuck all to do with it either - all the dog attacks this decade involved a dog known to the person (invariably their own dog or a family members) and it usually happens in the house. The only exception this decade was someone killed by their neighbours dogs which escaped - again, it wouldn't have been helped by dogs being on lead in the local park.