Because the problem with purebred dogs is that they're not inbred enough, let's start thinking in terms of "endangered" breeds that should be rescued by breeding lots and lots of dogs from just four ancestors.
I'm a serious dog people, as those who know me will attest. But breed-fetishization holds no appeal for me. We've been mixing dogs' genes for thousands, possibly tens of thousands, of years, to meet our needs and desires at the time. There's nothing sacred about the sub-speciation that happened to be in effect when the Kennel Clubs came into existence. The otterhounds are cute, sure. But if we've inbred the otterhound to the point of epilepsy, and we no longer need dogs to hunt otters, why on earth should we go on inbreeding them and trying to create demand for them where none exists, instead of reshuffling the genetic cards and getting some healthier mutts and, eventually, new breeds? The need to have dogs available as props for historical reenactors and cosplayers doesn't really strike me as compelling. If there are enough of the cosplayers to sustain demand, that's fine, but if there aren't, I don't think the extinction of the breed would be an object of great concern. (The life of each individual dog is an object of concern, but not the fate of the breed.) Dogs' genetic differentiation and specialization is a human creation for human needs; the otterhound- polar bear analogy doesn't hold. And we will, happily, go on having, and making, lots of different kinds of dogs for the foreseeable future.