The argument about what happens to puppies is dependent on whether you look at a specific puppy or pups in general. If that specific puppy isn't bought by the OP, several things could happen. The seller could put an advert on Gumtree, on the internet, give it to a friend, or worst case, dump it/shoot it/use it for bait in a dog fight. If it's a bitch, it could well end up in the hands of a puppy farmer, to continue the saga.
As with any market, the seller is responding to demand. If more people refused to buy from dodgy breeders, and went to reputable sensible breeders or to rescues, there would be fewer dogs in the system. Reputable breeders often do keep in touch with dogs they have bred, will take back the dog if there's a problem and will grill prospective owners very thoroughly indeed. Purchasers will often have to sign an agreement stating that they will not breed from the dog and commiting to return it in case of a problem. I haven't even touched on the issues around health testing - can't imagine many backyard breeders stretching to thorough and comprehensive pedigree examination, tests, record keeping etc.
By buying the puppy at the price demanded by the seller, the purchaser is sending a clear economic message - This works. The seller will, in economic terms, be perfectly sensible in then thinking - Right, I've just sold a load of cute pups at £400 a pop, thank you for that lovely tax free bonus. I'll do that again. It's interesting from an economic point of view because the externalities are not borne by the seller so they are effectively getting a free ride. In addition, many of these breeders are not declaring the income made so are making a very tidy tax free profit - I did some work on tracking one particular "hobby" breeder recently and they were making around £20,000 a year tax free income including sales of pups and stud fees. This makes me even angrier since undeclared income like this is money that is not going into the general economy to support things like hospitals and schools, but straight into the back pocket of a greedy, unscrupulous breeder.
I have no quarrel at all with reputable, responsible breeders (of whom there are several on MN) but too often people get involved in breeding for the wrong reasons, and make a mess we all have to pay for.
I am less familiar with the situation with cats - I imagine it is similar? 