It appears to me that the crux of the matter here is the degree to which one allows his/her analytical thought to be overridden by his/her emotions. Everybody has some level of emotional attachment to pets and livestock. Losing pets is difficult because we've allowed ourselves to develop emotional attachment. Allowing ourselves to build up that same level of emotional attachment to livestock is not appropriate, but an easy trap to fall into. It is then that we find ourselves unable to dispatch them for butchering. Look at the explanations given about becoming able to butcher once the livestock is no longer "cute" and cuddly. The only thing that has changed is our level of emotional attachment. If we look at it from a purely, analytically, rational point of view, we realize that it is more humane to the animal for it to be instantly rendered unconscious/brain dead by a blow to the head or cervical dislocation or gunshot than for the animal to die slowly from disease, injury, or old age. Even when we're dealing with a pet with which we have emotional attachment, don't we logical people prefer to have the animal humanely put down rather than allow it to suffer a prolonged terminal illness?
Those people who scream about animal rights and criticize raising livestock for food have surrendered all rational thought to the complete control by their emotions. They are incapable of holding a rational discussion because they cannot control their feelings. We all have feelings. Control of those feelings is what differentiates rational thought on one end of the spectrum from lunacy on the other. Fortunately, few people ever lose control to the point of lunacy. But there are a lot of points of various degrees of control, or loss of control along that spectrum.