Sadly, some people seem to want a unique designer dog and the fancy name to go with it. These are the people who don't care about the parents (welfare, health & early life of pup). Some unscrupulous people buy into this fashion and supply to desired item.... sorry puppy.
Equally, I think it's impossible to police dog breeding. If you start licensing breeders and charge them for the privilege, I fear, that we would loose a lot of good healthy breeding stock in this country. Resulting in a small gene pool and giving rise to more health problems through inbreeding. (Because that is why we need health testing in the first place.)
Crooks will always be crooks and go "underground".
Supply would outstrip demand very quickly (as it already has, hence breeders with waiting lists). This will result in more import puppies, with more health problems, spreading diseases like rabies. (I know, this is already happening on a smallish scale!)
The Kennel Club is not an government organization, it's a private entity, designed to keep records of dogs, setting breed standards and organize dog shows (all kinds). So not sure why/ how they want to police "illicit dog production".
I think sometimes, NM are very harsh on "small scale breeders". Calling them all sorts of names and greedy. In fact, these breeders have an important role in keeping a healthy gene pool for their breed. They may not show or do doggy sports, and the mum is just a pet. But, please do your math, out of 10 puppies, you might get one show quality. And if you take 10 show quality puppies, only one can win 1st prize.
They, if done properly, provide good quality family pets. As the puppies will be raised in a family environment often with children, they will be exposed to people from an early age.
So, unfortunately it's up to the puppy buyers to do their research, know a bit about dogs and only buy puppy if 100% comfortable.