Cats and dogs, however, are so common that animal shelters are overwhelmed.

In the USA, this has not been true for a while. In fact, there are charities specialized in just shipping dogs from the American South, where they still tend to have pet surpluses, to the North. (I was stuck behind one of their 16-wheeler trucks a few years ago, and looked them up on my phone.) Here's someone arguing we don't have a deluge of unwanted animals, but a shortage: "How 'The Pet Revolution' Unleashed A New Top Dog In America"

Despite those depressing ads featuring Sarah McLachlan's "Angel," Cushing claims there is now a dog shortage. "There is a deficit of two million dogs per year, and it's growing," he writes. Cushing's estimate is based on a year-long national survey of 1500 American households, which asks them how many dogs they currently have and where they got them from. His estimate is also based on assumptions about how many dogs Americans will want as dogs die and the human population grows.

He sees evidence of the shortage in prices and rates of adoption at shelters. Southern and Midwest states, he says, spay and neuter their dogs at a lower rate, and as a result, we've seen the growth of "canine freedom trains" from shelters there to homes in other parts of the country where there's the most chronic shortage. But, he says, those shelters can't keep up with the demand. And that worries him. "If the demand for dogs continues to exceed the supply, dogs will become a luxury item," he writes. Cushing wants to find a middle ground in an ongoing war between big breeding operations and advocacy groups, like the Humane Society and the ASPCA, over mass production of puppies. He argues we can allow large-scale commercial breeding while also making sure breeders rear puppies in a safe and healthy environment. He also suggests that maybe we scale back spaying and neutering. But everybody's got a dog in this fight, so the politics are ruff.

The lack of pets leads to some perverse consequences - in a beautiful example of Shirky's law, we have dog rescuers manufacturing the problems they take donations to solve: "Dog Fight: Dog rescuers, flush with donations, buy animals from the breeders they scorn". (I assume this is also what happens with the popular 'fox rescue' people when they purchase foxes from fur farms, not to mention completely fake animal rescuers; in general, see "Buddhist Animal Release Practices: Historic, Environmental, Public Health And Economic Concerns", Shiu & Stokes 2008). And of course, during the coronavirus pandemic, pet supply dried up as the demand went through the roof, eliminating any last vestiges of 'overwhelmedness'.

This shortage extends to pedigree animals too; some of my relatives are dog trainers/showers, and have discovered themselves unable to even contract for dogs, but forced into what I can only describe as puppy futures and breeding out their current bitches to try to get new puppies.

Eat the cute animals?! Absurdly wasteful. There are too many people who want to eat them with their eyes.

EDIT: I understood your point perfectly. I have nothing against eating cats and dogs. Aside from it being a bad idea for reasons I just outlined.

I've eaten dog meat, and it wasn't nearly as tasty as pork or beef, or even chicken.  I'd expect cats (as carnivores) to be pretty bad as well.  I do enjoy eating lamb, and I like seeing rabbit on the menu (though it's not my favorite, texture-wise) - is that cute enough for you?

edited to add some meta-clarity.

Note: I was trying to criticize the hypocrisy of people who treat farm animals in ways they would never treat pets. It seems people missed this.

Yes, I don't think anyone missed this.  It wasn't unclear, just unhelpful and unwelcome in this forum.  Just pointing out Other People's Irrationality is not instructive nor interesting.  If it's something you've seen here, then it's fine - you just need to to identify what specifically you're objecting to.  If it's not something that many of us seem to support, you need to do a lot more analysis and exposition of why it's a useful thing to post here.  Perhaps try to deconstruct WHY someone might hold this view, and give some examples of whether your tactic is effective or not.

People (including me) responding at the object level were presumably as unhelpful to you as you were to us.  What would you change to make either side of these interactions more useful or entertaining?  

Note: I am an unabashed carnivore, and a non-universalist - I assign many orders of magnitude different values to different entities, and don't expect that my assignments match anyone else's.  Unless you can address this crux, you're unlikely to convince me that the difference between non-existence and brief, non-compassionate existence at a farm for a few dozen chickens a year (my marginal impact) should outweigh my pleasure and convenience.