I think very few "absolute" statements are true, life is full of all kinds of choices some better then others. It is true that caring for chickens and other farm animals brings with it all kinds of different situations then typical pet animals. With chickens especially it is hard to find vetrinary services to "do the dirty work" of caring for an animal, that is why you see so many threads of performing invasive medical treatments and repairs to sick and injured chickens. Chickens do get sick worse and injured more often then a typical, say, cat or dog, requiring interventions some mild like giving meds, others more severe including stitches, and euthanizing. How you choose to euthanize(or provide invasive care) is still a choice YOU must make, whether it is you doing it, or a friend or relative that has agreed to help you with this, or finding a vet willing to take care of the chicken, or putting the chicken in a quite safe place and letting "nature take it's course". All these choices have good and bad consequences, some are better then others, some make us squirm or cringe. Do YOU have to be able to personally physically kill an animal to own one, no, there are choices. Should you have a plan in place for it, yes. B/c it WILL happen it is not a matter of IF but WHEN.
A lot of us thought we couldn't "do it" until we were faced w/ our first mortally wounded chicken that we raised and loved and to know how much pain it was in and know it was the only way to end it's suffering and couldn't bear the thought of it suffering anymore more then our personal discomfort. Have a plan in place before it happens b/c you won't be in a position to rationally make a plan during the crisis.