The "idea of purebred dogs" is maximization of desirable phenotypic traits, and is identical to the idea of artificial selection throughout agriculture and animal husbandry, which has underpinned the sustenance of our civilization for many thousands of years.

The idea that artificial selection is a terrible idea, on the other hand, is a fairly recent one, and it is entirely informed by its political dimension of eugenics as a peculiar form of totalitarian intervention into human life. Inasmuch as it's terrible to preserve purity of dog breeds, this is entirely explained by a problematic breed standard, usually because it emphasizes some aesthetic feature over generic vitality and ability.

There's another terrible (and wrong) idea here, namely that the best way to maximize those generically good traits is through random interbreeding and preferring crosses with high phenotypic and genotypic divergence; that "mutts are the smartest and healthiest dogs" and so on. This, too, is overwhelmingly informed by a misguided generalization of opposition to a certain abhorrent political project. But reversed stupidity is not intelligence, and misguided anti-nazism is not a good rule of thumb for animal husbandry. In reality, it's not a surprise that e.g. the best, kindest and smartest assistance dogs come from breeds we know as reliably helpful.[1]

It's also not a surprise that Labrador Retriever dogs have unreasonable appetite and high obesity rate: we know the exact specific gene that's broken which explains it. I hope we fix it one day, that's the least they deserve. We easily can fix it, too. And this is the state of health issues with most non-decorative breeds.

> We understand the ramifications of inbreeding in humans

When we understand something, we usually can quantify its effects. As it happens, we do understand inbreeding and why it's bad, both in humans and in the general case. It's mostly explained by increased occurrence of homozygous-recessive phenotypes. Basically, most deviations from evolutionarily optimized alleles are suboptimal, recessive, and in the limit, get weeded out by selective pressure (purifying selection). But a very limited gene pool can become saturated with recessive alleles, leading to many specimens who are homozygous-recessive and get the suboptimal trait; and in bad cases, a recessive allele reaches fixation.

Accordingly we have different relevant metrics and notions, e.g. of effective population size [2] (number of effectively distinct breeding units in the mathematical model of this population; more relevant than census size when you have technically large but highly inbred populations – corn, dairy cattle, pugs, Austrian monarchs…) and of minimal viable population [3] (number of specimens sufficient to ward off the drive to homozygosity and eventual inbreeding depression). The typically given safe number is about 500.

Even very small natural human populations (on the scale of tens of thousands of people, a minor ethnicity or tribe) do not have any inbreeding problems, nor get any heterosis aka "hybrid vigor" i.e. benefit from interbreeding and canceling the accumulated inbreeding depression. Observations to the contrary come from extremely bad cases, like people from isolated premodern villages with some 5-10 families moving to the city for the first time in centuries/ever, and their children being markedly healthier in all sorts of ways (that said, even in such cases, improvements in nutrition, education etc. can explain an obscene share of the effect). But these children do not get more (or less) fit grandchildren after they marry even more genetically distant people from another country – they're already free from the curse of inbreeding depression their ancestral populations carried. What drives these effects is heterozygosity, not total diversity.

This is true for reasonably maintained (i.e. without severe recent bottlenecks) non-decorative dog breeds as well. They don't produce super-great mutts. They're fine as they are.

1. https://eu.usatoday.com/story/news/nation/2018/12/18/griffin...
2. https://en.wikipedia.org/wiki/Effective_population_size
3. https://en.wikipedia.org/wiki/Minimum_viable_population