Not entirely on-topic, but can I ask about "co-owns"? Is this a specific, unusual situation you have (I've heard of people doing that for dogs that the breeder intended to use for breeding), or was this presented as the standard contract by the breeders? Because if so, that gets a huge "back away slowly" from me.
It is a thread digression, but I'm sure those not interested will skip this and forgive us. Yes, in purebred dogs it's pretty common that breeders want to co-own puppies they sell. The good thing about ethical breeders of quality dogs is that they feel a responsibility for any puppy they've bred for its whole life. If necessary they'll take back a 9-year-old spayed or neutered dog they bred and give it a good home. The flip side of that is they want some control. They don't want a puppy buyer breeding a dog they got as pet quality and swore to never breed. They do want a puppy buyer who promised to show the dog, breed it and give the original breeder a puppy out of the first litter to keep that promise. I know of one breeder in this area who used her contract to get back a dog at 3 or 4 because the buyer left him chained in the yard and neglected him. So some breeders try to control what happens to their puppies with co-ownerships. Others use written contracts, and some use both. And I've heard of ugly disputes both because of co-ownerships and contracts, even occasional lawsuits.
Are you in the US? My dad has had purebred dogs my whole life, and he's never been asked to sign any kind of co-ownership agreement, nor would he. Nor would I. There's simply no way I would allow anyone else to have legal ownership rights to a member of my family (or even an expensive piece of my property). I don't mean to sound like I'm criticizing you at all, but I do find it an outrageous thing for any breeder to require. That is just such a hard no from me, as someone used to the idea of purebred dogs, that I would honestly be offended if I was presented with a contract like that. I would sooner buy a random puppy from the neighborhood.
I definitely agree that a reputable breeder will take a dog back if there's a problem. This is also something I've been used to from seeing my dad's experience with breeders. The one he's gotten his past several dogs from has a policy of taking any of their dogs back at any age for any reason. A good, reputable breeder cares about their dogs and wants them to have a good home, no matter its age. (My brother, years ago, bought a dog from what seemed like a reputable breeder. After several years, his life situation changed and he felt he had to get rid of it. The breeder was surprised by the very notion that he might take the dog back. My brother was stuck giving it to some stranger who didn't even end up being a good fit. My brother didn't want to deal with it. We have no idea what ultimately happened to the dog. So, yeah ... if you're gonna pay the money to buy from a breeder, I'd call a solid return policy an absolute must.)
I've heard of spay/neuter agreements, where breeders require that a dog be spayed/neutered so that they don't get used for competing breeding, which makes sense to me. I can understand where that is coming from, and while I don't love it (I do tend to want to actually fully own things I buy), it wouldn't be an issue for me since I would never deliberately breed one of my pets (it's shockingly dangerous for females and so often goes poorly). That is absolutely possible to do without the breeder having to retain ownership of the dog. (Especially if they just go ahead and get it fixed before selling. My cat was actually spayed at like 7 1/2 weeks because the rescue required it, and she was totally fine.)
I would never sign a contract where the other party tried to assure me that they'd never use it. "If you don't intend to use it, why do you need it?" and "If you don't trust me to not be a criminal, don't sell me your dog. If you do your due diligence and think I'm a responsible person, sell me the dog. You don't get it both ways." (Really, the same kind of contract advice that, as you say, Kris Rusch talks about for writing. At least the first point there, though the second is a bit like the "morality clauses" she's mentioned in recent contracts.)
Honestly, it's probably a moot point as far as my personal future, since cats just fit so much better into my life. But still, if I ever did decide to get a dog and wanted to get it from a proper breeder, I would still never sign a contract like that. Not all breeders use them, so there's no reason for buyers to accept them (unless their very specific circumstances make them make sense).
It does make sense in your situation, though, if you're old enough without anyone else lined up to care for the dog if you die. I can see that. It can be challenging enough for the family to figure out what to do with a dog; it would be even worse if the owner didn't have family or family who cared enough to find a good home. (Though I'd think another option, rather than co-ownership, would be to just include instructions in your trust after you make arrangements with someone, to cover any potential legal issues.)