The KC do have a similar system. The breeder can place endorsements on the dog's pedigree which prevent any progeny being registered with the KC. Sadly, just like with pedigree cats, the system is open to abuse and some people will breed anyway and just not register the progeny, as they can't. Hence we see all the ads which say 'parents registered, but not the pups/kittens'.
I breed Golden Retrievers and place endorsments on the papers of all puppies I breed. The buyer them signs a contract, part of which expains what the endorsements are, why I have decded to place them, and, that if they want to breed in the future, how to get me to lift them. I would lift the endorsements (only the breeder can do this and it means progeny can then e KC registered) subject to good enough health tests results, the dog or bitch being of a suitable age, temprement and a good enough example of the breed. I am maybe not as strict about lifting endorsements as some breeders are, as some will only lift if the dog has an extremely successful showing career or is a SH CH. Personally, I think if people really want to, they'll breed anyway, and I'd rather they did it the right way wth my help and guidance than just thought they might as well not bother to health test and do it anyway. I try and maintain a good relationship with all my puppy buyers, so they will come to me if thy want to breed.
Neutering with dogs is not something I would insist on. It is essential for cats, or at least queens, that will not be bred for health reasons, but it's not so cut and dried for dogs, and because of that, neuering is not something I would ever insist anyone did.
A lot of the problem regarding dogs in the UK is caused by lack of knowledge on the buyers part, or even selfishness. For example, Joe Smith works full time and so doshis wife,so no decent breeder will sel him a pup. He thinks it's his right to have one though, so goes to a puppy farmer or BYB who couldn't care less once they have the cash. OR, people don't want to wait, and the fact is, most decent dog breeders do not breed often and people will have to wait. Not everyone is aware of the importance of health tesing either and even if they do, some people think it won't happen to them. I do think education would go a long way to help.
I've gone on abit, but I aso have a Ragdoll and they are fab cats.