I could write a whole book on why cross dogs but I will just provide 5 points.
1) purebred as conceived of today, is a great restriction of the gene pool compared to the past. It means being restricted to a breed registry that represents sometimes, an extremely small population. This phenomenon is relatively recent, especially in the LGD breeds, having begun in many cases as recent as the 1950s or 60's or even much more recently. This is but a blip in time in the history of the LGD.
2) most breeds as defined by breed clubs today, were defined by political boundaries that did not exist 50 to 100 years ago. Shepherds were migratory people, and their dogs, and dog breeding (of which their was little control except that one would hope they mated with other shepherd's dogs on the mountain passes or villages) covered a very large area that extends well beyond today's political boundaries in Europe. I should point out that in Eastern Europe for example (Poland, Hungary, Czechoslovakia) what was basically one 'breed' type became 3 or 4 different breeds in modern times. So instead of interbreeding as they once did (and sharing a much larger gene pool) these breeds have become very narrow in their gene pool only just recently. Meanwhile the dogs now defined as a separate breed are merely close cousins.
3) selection of the next generation did not occur so much at mating, but at whelping where pups were selected on preferred coat color, often only two puppies were selected. Thus new genes crept into the LGD base on a regular basis.
4) A good read is Ray and Lorna Coppinger's "DOGS" a book that goes into the origins of the domestic dog, including a great deal of discussion on the evolution of behavioral traits. In that book Ray puts forth the theory that a LGD is what it is, due to a history of crossbreeding that has kept the LGD gene pool largely more diverse than what we define as a 'purebred' today. His opinion is that the hybridization of the gene pool might actually be required to maintain certain LGD behavioral traits.
5) Independent research by Ray and Lorna Coppinger, and by Jefferey Green, where the performance of various breeds, including hybrids were compared found that the hybrids scored higher and were more successful than the purebreds.
Now having said all of that, if I want a herding dog, I want a purebred border collie. But a border collie does a very different job from the LGD. So what is good for the border collie, where it is ok to have some rather obsessive behaviors, is not necessarily good for the LGD, where obsessive behaviors can become quite destructive.
Hybridization (the crossing of two LGD breeds) in the USA has been mostly a practical matter. If you raise Pyrs, finding a purebred Pyr might not be too difficult, but what if a purebred Pyr does not really suit your situation? The Polish Tatra suited me very well and for 9 years I had an excellent Tatra male here. He had everything I wanted, the picture of health, assertive with predators, yet gentle with stock, and stayed with his flock. I spent all of that time, and $6K trying to find a suitable Tatra female for him. All four females failed to meet my criteria (either due to joint issues or behavioral issues) and were turned into pets without producing any pups. Meanwhile Koci produced 150 hybrid puppies that have been excellent working dogs throughout the country. What a waste it would have been to not use him because someone thought crossbreeding was a bad idea, or worse, what a bad idea it would have been to perpetuate the bad traits in those four females in the name of breeding purebreds! If you have a pair of good dogs....and they are truly good, use them! whether or not they are the same breed.
Lastly... This is the new world with a new environment and different needs than the countries of origin... The quarter horse was born out of necessity by crossing breeds much as I just described above. It was a practical matter. What we have brewing here in the US is perhaps a new breed, the success of which will be fueled by the much larger population and greater selection intensity available because it is not restricted by the books, the fads, and the fetishes, of a breed club.
Tamarack Prolific and Ile de France crosses