That's why they're awesome. Cats are like poetry--a large part of their beauty lies in the fact that by traditional standards of utility, they're pretty useless. But as models of evolution, they're perhaps the most successful domesticated animal, depending on what you value.
Success is a relative term--cows are incredibly successful as a group, but they had to give up so much of their autonomy to hook up with humans that while they live in far larger numbers as domesticated animals than they would have had they stayed wild, they also serve as food stock and often live in deplorable conditions. That's their tradeoff. Same for any animal that humans use as food.
Then there are the kinds of animals that depend on human settlements for survival, but which have never given up their autonomy and become pets--pigeons, crows, those sorts of things. They provide a service--they clean up leftover food and the like--but they also are on the butt end of a lot of human disdain and in many cases, humans try to wipe them out, or at least control their numbers. Their lives don't have the sort of ease that an animal that's become a pet can generally aspire to.
Which brings us to cats and dogs, and I'm going to argue that cats are the more successful of the two. Both groups have largely managed to worm their way into the interior of human life, which means they get the benefits of living close to humans--a secure food source, relatively better sanitation, treatment for injuries and parasites (depending on the owner, of course), and a generally higher standard of living and extended life span--but without having to serve as a food source for said humans (in the US at least). We've even gone so far as to provide them protections against abuse.
Of the two, dogs--at least certain breeds of dog--are expected to be utile, usually as hunters or guards. That this is becoming less the case is a sign, to me, that dogs are becoming more like cats, especially when it comes to the smaller, more frou-frou breeds. After all, is a Pomeranian really a useful animal? Seems more decorative to me.
The real argument to me, it seems, depends on whether you think there's a disconnect between loyalty and independence. Dogs are, in my experience, exuberant about their loyalty, and I think that's a big part of what dog lovers respond to. Cats aren't generally as exuberant, though I would argue that they can be just as loyal, when they're given a reason to be. It's the second half of that equation that I find to be the most interesting--cats, especially indoor cats, haven't given up as much of their autonomy, but they've gotten all the benefits of being close to humans. And they don't really have to give anything in return--they're not expected to do guard duty or flush quail. They're like poems--they're just expected to be, and every so often, they evoke an emotional response in you, good, bad, confused or otherwise.