One of the more galling rationalizations for killing a pet whose near future looks dark is that, in some way, the animal is ready for death and perhaps even sending a signal to the owner that it's time to go and please do the job for me.
Their movement gets stiff, they don't walk or jump around as much or at all, they aren't as playful, they stop eating and drinking — they're just laying around, waiting for you to put them out of their listless misery. It's only humane of you to honor their request and have a doctor come pump barbiturates into their veins to shut off their nervous system.
Wait just a second — if they're so miserable and beyond all hope, why aren't they trying to do the job themselves? Not like it would be hard for most pets. Over a lifetime, they have cultivated a good sense of what things and places are dangerous, so instead of avoiding them, they could go toward them. Y'know, walk out into busy traffic, climb someplace high and go splat on the ground below, pick a fight with a nasty predator or rival... anything, really.
In fact, animals never commit suicide (which is not to say that they don't sometimes behave in a way that results in their own death). Their survival instinct is too strong. I suspect the same is true for human beings, and that suicide is a maladaptive disease of civilization (the exact mechanism being irrelevant for now). I don't recall ever hearing of hunter-gatherers taking their own lives. In any case, we sure don't see it in animals, particularly not pets who are winding down their last days.
Laying still, breathing slow, refusing food and water is their way of conserving rather than wasting energy so that they'll live as long as possible in the final stage of life. Yes, even trying to eat, digest, and excrete food would be a waste if the bodily systems required are performing poorly or winding down.
If they seem sad, it is because they sense the end is near — not sad because you're taking too long to bump them off already. Like any other living creature, they want to go out on their own terms, and not have their lives taken from them by some paternalistic authority. They want to be respected and leave with their dignity intact, not to get snuffed out in a final disrespectful humiliation.