It's a bit like humans having a genetic test to deter main there offsprings failings - we wouldn't allow it - so why do we do it for dogs?
Humans can indeed opt to have genetic testing and counselling before trying for children if there's a risk both could be carriers for various conditions.
When it comes to health testing dogs it's not about avoiding minor "failings", it's about avoiding disabling or even life-limiting health problems.
Whoever bred DDog2 (she was a rescue so we've no idea) didn't bother health testing her parents. Unluckily for her both parents were carriers for hereditary cataracts and even more unluckily for her she inherited the gene from both parents. She started losing her sight when she was barely more than a puppy and was completely blind by 18 months old. She had corrective surgery (to the tune of £5000) which was moderately successful but she still has far less than perfect sight.
I've no idea if any of her siblings suffered the same but each will have had a 25% chance of inheriting both genes and being affected. It would have cost less than £100 (£48 per parent dog) to find out that both parents were carriers and thus shouldn't be bred together.
She's also got terrible conformation (poor thing, she's like the poster child for crap breeding) which has contributed to her developing IVDD at a relatively young age. Obviously there are no simple DNA tests for good or bad conformation but if you breed two dogs together who each have good conformation and whose conformations compliment each other then you've got a pretty good chance of producing puppies who also have good conformation. Unfortunately it also works the other way and poor conformation can have a real effect on the health and fitness of a dog.
I don't see crossbreeding as inherently problematic, I don't always think there's much point to it (it seems a lot of "cockapoo" owners want something that's remarkably similar to a miniature poodle without the show clip for example), but it's not something I can get worked up about as long as it's being done with care.
Shitty breeders are the real issues and they exist in the worlds of pedigrees, crosses and mongrels alike. At the moment it just so happens that crosses (seemingly the more random the better) and French Bulldogs are ridiculously popular and so there are endless puppy farms, BYBs and misguided pet owners who think Fluffy would be the perfect mum churning out puppies to meet the demand with very little care being given to health, conformation and temperament.
I think the popularity explosion of crossbreeds partly came about because of the belief that pedigrees are all riddled with disease and crosses or mongrels are guaranteed to be healthier. Of course it's not as simple as that but it's a myth that still perpetuates.
There needs to be a huge culture shift when it comes to both dog breeding and puppy buying, particularly the latter. As long as people buy puppies from puppy farms, BYBs, puppy dealers, pet shops and so on they will continue to exist. Unfortunately I can't really foresee a time when there won't be people willing to compromise on where a puppy comes from just so they can get the puppy they want, when they want regardless of whether they've picked an appropriate breed or whether they are actually suitable dog owners.
As for the actual question the OP asked; I think the vast majority of breeds have a dedicated enough following to ensure their survival. I agree it would be a huge shame to lose the breeds we currently have, or for those breeds to end up as generic dogs just in different looking suits. Cross breeding isn't the only danger facing pedigree breeds though, closed gene pools inevitably spell disaster in the long run. I'd love to see the KC introduce things like working tests (where appropriate), temperament tests and approve sensible outcrossing between similar breeds to help improve genetic diversity.