The other dog in the picture (Rhodesian Ridgeback/Lab cross) is a different story. Rescued from an abusive home you can tell she's a bit broken; a lot better but older and still quite grumpy. She has taught me one thing - I'd never get another rescue dog, ever.
So I may appear hypocritical with our dogs and having one that's been obviously affected by bad nurture in it's past and arguing the killer instinct is in the breed etc, but I still stand by that. Good nurturing glosses over bad breeds' instincts but they are still there under the surface and in some cases it doesn't take much for that veneer to be removed.
By contrast, I'd always look to a rescue centre - All three of mine are rescues, two of which were clearly very abused mentally and physically, yet all of them have responded amazingly to being homed with humans who simply don't treat them like ****. Each has their own quirks of behaviour, none of which are associated with breed but which aren't detrimental to anyone either. I swore we'd never have a Collie, because they're all mad and too energetic... yet one down the rescue home was so un-Collie-like, she was perfect for us and far better behaved than most other dogs.
Breed has very little bearing on individual personality and behaviour.
Seems to me the obvious solutions to this are 3-fold.
All breeders require certification and registration, in an effort to prevent "back-garden breeders"
All dog owners require a license and registration of both themselves and their pet, with heavy criminal penalties for those who do not.
All dogs must remain on a lead / harness at ALL TIMES in a public space and those stupid "extend-a-leads" should be banned outright.
After all, it is the minimum requirement(s) we have for owning / using most other things in a public place which are considered "potentially dangerous"
Breeders tend to stick to specific purebreeds and 'designer' crossbreeds like Cockapoos.
Mongrel dogs tend to be better off, as being less 'inbred' means fewer health issues, but they're not big moneymakers, so breeders of them are generally limited to amateurs and one-off circumstances for teh sociable ones, and back-street puppy farmers for the nasty ones. Neither of them will be interested in licencing.
Owner licence & registration is a lovely idea, but licencing was scrapped yonks ago because so many people just ignored them, and would likely never work now because it'd require a massive force to enforce it... and registration would still get ignored by the scum.
On lead at all times also is never going to work, as many dogs need to actually RUN for proper exercise and no human can match them. I take mine to large publicly accessible places so they can run, albeit mostly at night when no-one else will be around to bother us, but partly because all the enclosed places (of which there are too few) charge insane money.
I agree about the extending leads, though!!
But most importantly - Yes we have all these as minimum requirements for other such things, yet we still have a great many offenders. About 2½ million people are prosecuted for motoring offenses each year, and that's just the few who are stupid enough to get caught, because there is a large force looking out for such issues.
You might have some luck by working on cultural change, to make ownership of certain dogs a very unfashionable thing (ideally starting with Pugs and all the other badly mutated/mutilated breeds that suffer simply from how they were bred), to negate the status of the status dogs that the scum love so much.
You
may also get some luck from increasing the powers of the RSPCA and giving them some Police-like training to deal more heavily with bad owners.