That's why only the best of the best must be bred. Dogs with titles and OFA and CERF certifications by licensed veterinarians (not "health checks" but actual exrays etc). If people didn't breed dogs willy nilly a lot of the health and structure problems (not to mention temperament problems) in dogs today would not be as serious as they are.
Mixed breed dogs often get the short end of both sticks. My own dog Bella is a boxer mix who at 3 years old had life threatening surgery to fix a ruptured disc in her spine that left her paralyzed. This sort of thing doesn't usually happen in boxers or dogs that young but because of her mix she wound up with a myriad of health problems from IVDD (intervertebral disc disease) to hip dysplasia and a 100% chance of having arthritis when she gets older. Most mutts out there are not as healthy as people would have them believe.
AND it's not like breeding is necessary to get healthy dogs, there are some very healthy animals out there(and we generally don't have a 'wild caught' problem with dogs) They are domesticated animals, not being taken from the wild.
Some animals should be bought from breeders, some you shouldn't. It's considered better for reptiles in general if they are bought from breeders--pet stores usually have wildcaught specimens who are taken by people who don't care for their health. They then even overcharge for them(leopard gecko for 45 dollars, a healthy normal morph baby from a breeder is more like 10 dollars) and then expect them to die anyway!
Same with birds, it's not a good idea to buy a bird from a big chain petstore, if at all possible, get one from a breeder. You can get them handfed, loved and played with, and it is not a scared wild animal who feels trapped now that it isn't outside!
I have a purebred dog, I had a pure bred cat, but I love getting animals from shelters too. Honestly though, there is a myth that 'mutts' are healthier than purebreds--while this CAN be true, it can also be VERY false!! We had a mutt who seemed to have every health problem that each of his parents had. He had seizures like his boxer mom, bone issues like his golden retriever father.. It was a very difficult dog to take care of, but I loved him. There actually are some purebred dogs with very few if any natural health problems--the closer they are to the original 'dog', the healthier. Also seems the ones less popular(not bred for looks) are the ones that are healthiest... Maybe because the only people breeding them do it for the breed, not to make money?
Edit: Oh! It seems 1fish2fish and I agree on the dog issue!
On the note of wild and domesticated breeding, being in the military I have unfortunatly seen people do just that. The animals do turn out beautiful but there are so may risk involved such as pack instict vs loner, aggresive vs docile, and genetic disorders.
I love most dogs anyway, my inlaws have a dautson thats prone to seizures every couple months and a mutt who is just kinda of dumb(not trying to be mean its just the truth, we think hes rott, beagle, and bluetick mix). My wife and I have a chiuaha that we saved from a bad situation that sounds like he has ashtmah when he drinks to fast or around certain perfumes or plants, and a mutt who is extremely kid friendly and smart but still protective(expecially of my 2yr old, we think hes beagle and husky), my wife found him at the pound when he was just a pup and couldnt resist him.
We had a mixed breed dog who was the best dog we ever had, He was pretty healthy until my parents split up and my alcoholic father took him and didn't take care of him. Left him out in sub zero westher, fed him nothing but fat off of steaks and other meat. He was such a loving dog,very loyal and very protective.