No Elitegaurdianpresa, unfortunately not, it's also not even a matter of opinion. The first dogs shows held were never meant to access working ability.
There were no such "dog shows" as such. Just dogs proving themselves in the field, thats how they were selected. Dog shows are a Victorian invention along with closed stud books. The purposes of these shows was never to access the working ability of the dog and they still aren't today. It all began around 1859 in the UK. Yes most were working dogs shown at first. Not function tested, "shown", entirely judged on conformation. It's not that long ago either.
Registries were closed and the obsession with purity was born, not in the field, yes right in the show-ring.
Line breeding is synonymous for inbreeding. It eventually catches up with you and you have to outcross within the breed. But the outcrossing options become more and more limited as everyone is doing the same to popular sires that win ribbons, the dogs winning best of breed at any of the top dog shows today. They don't say "stuff you" can persist with their lines of non-winning dogs do they? Winning at shows is their entire point in breeding.
Breeding work to work was and is still the only way you can hope to maintain a working "line". Not line breeding.
Breeding only working dogs proven in the field, like to like is the way. Dogs that don't make the grade in the field were not bred with. This has been the time honoured method of breeding working dogs. It's predictable. The critical point here is most truly working dogs were never bred under a closed stud book like show dogs. Breeding working dogs also always relied heavily on tweaking the dogs used, be it for size, temperament, tenacity, recall even in some instances colour, any number of desirable working traits.........not within a closed registry. Thats to say freely outcrossing to different breeds and types, where and when needed to maintain a working line. This is not possible in a closed registry of any type of dog, working or purely show dog. This way even if there was a small amount of inbreeding, breed types remained geneticly robust.
Most dog, non primitive, breeds today are of course in fact as a result made up of many different "types" or breeds, selected over the decades even centuries.
Even companion breeds were selected to do a working job. That is until dog shows and closed registries were invented. A mad insane eugenicists fixation with "purity" evolved, was sanctioned and you had the birth of Crufts. As dogs became redundant in the field in various functions and pursuits they found them selves entirely bred along the showing principles and lost all working ability they ever had. Countless breeds fall into this category today, almost 95% of them. From the Great-Dane to the Corgi. Also being closed registries mean they were a shrinking gene pool from the very start of breeding in closed registries. In just over a hundred years many breeds have been destroyed. They also completely changed phenotype as a result of selecting for extremes that win in the show ring. Even if they still had any drive or ability they have morphed completely out of the "ball park". Functionless other than sickly show animals. This is also valid for other livestock species where showing took precedent over function. A show bred dachshund can't even get down a fox hole, its chest is too deep, its legs are too shorts its body too long.
Is this important, well not really because there are types that are bred that can still get down fox holes. Is it important most Dachshunds are kept for companionship rather than hunting (BTW there are still strains of hunting teckles) no. But they must be healthy and functional as dogs at the very least to even function as pets. Sadly they aren't and suffer from the same problems (many also unique of course) that most pedigree dogs suffer from. Genetic diseases and ailments, many life threatening caused by their extreme phenotype. We still need to select for function and tweak genetic vigour with outcrossing to other breeds or types to maintain any breed.
Absolutely, outcrossing to different types/breeds has always preserved the function of working types.
Many livestock guardian breeds never had pedigrees never mind closed registries. Dogs from different regions regularly met up and shared their genes. Where they didn't they became inbred weak and died out.
Pedigrees in this day and age where we are no longer nomads living in the hills are though a useful tool to avoid inbreeding. BTW you can also tweak and even fix phenotype by never line-breeding ever.
I think you mean there is "no" mathematical equation to predict which genetic traits will be passed onto off spring"? Only by appraising a certain percentage of off-spring for function and dominance of any inheritable traits, can you tell. Prepotency does exist. For some diseases where the genes responsible are known there is also an equation of how best to avoid. But if your referring to "Lloyd Brackett's" line-breeding formula, a denialist of Mendelian genetics on par with the anti-science creationists view of evolution, the same who did more to destroy pedigree dogs including his own breed, the German Shepherd than perhaps anyone in history you would also be wrong. All he could safely predict was the inevitable "severe and intentional degradation of both health and conformation" in dog breeding.