I struggle to find the concept of eugenics categorically applicable to this case.

I'm not sure what's confusing about it. A species represents a line of genetics, and you clearly see yourself as a moral authority on what genetics do and don't get to continue to reproduce themselves. For the record, I don't necessarily disagree with this on principle - negative genetic traits that cause pain in animals can and should be bred out of them - I just believe that the type of eugenics you are invoking is a morally unjustifiable type.

what really is the material reason for keeping in existence animals who have been selectively bred solely for selfish human means?

There is no "material" reason. It should be done because it is the right thing to do. If you want a moral justification, I would say that life is valuable in all of its forms, and that domesticated animals represent just as valid of a form of life as wild ones. Their existence makes the world a better place and extinguishing them would be morally reprehensible no matter how nice you are about it or how slowly you do it, so establishing a new paradigm where they continue to exist in a way that balances environmental costs and ethical concerns is better than the alternative of sterilizing them and letting them all die out.

I believe that you have succumbed to a kind of "animal ownership realism", where you can only imagine animals living alongside humans through the lens of exploitation that they are currently subject to. What I would advocate is an advancement of humanity's relationship with its domesticated animals to a kind of non-exploitative symbiosis. Dogs cannot live in the wild, and that's okay. Cats are specialized to live in human domiciles, and that's okay. There is no reason to assume that we cannot still share companionship with dogs or live alongside cats or any of the hundreds of other human-pet relationships in a future that has abolished all forms of exploitation, human or otherwise.

I am vegan. I have been a vegan for over a quarter of a century now. I also have a cat. There was of course breeding of his ancestors for exploitational purposes, but there was also co-evolution that benefited both species. Regardless, my relationship with him is not one based on exploitation but on genuine mutual companionship. Our relationship is mutually beneficial on a number of levels. I am confident in saying that I not only allow him to live an enjoyable, self-actualized cat life, but that I also at times bring him joy. And he certainly brings me joy. I respect him as an individual being with his own wants and desires and do my best to fulfill them. I know of plenty of other humans who have similarly mutually beneficial relationships with their nonhuman companions. This is not welfarism and it is no more exploitational than many human relationships where a power imbalance exists by necessity but is still one based on mutual caring. I think the welfarism argument also completely ignores the reality of co-evolution (as seen between many other species that don't involve humans at all). Humans and nonhuman domesticated companions can and frequently do exist in non exploitational harmony. There is no reason this should be at odds with veganism and no reason we need to move towards phasing out the domestic cat (or dogs, etc.) as a species in order to be morally, ethically, or ideologically consistent as vegans.