Avada Kedavra is the plot element which both distinguishes the main character and serves as the signature attack for his antagonist. An attack that is (by definition) as terrible as Voldemort himself.
Harry does not show any particular gift or skill at wizardry, nor a particular amount of intelligence or even some kind of shrewdness. The most basic and obvious petty tricks (household magic, everyday magic devices) which every young wizard uses without wasting a thought still surprise him even after 3 years at wizardry school. He also isn't particularly good looking or athletic, his only notable ability is flying a broomstick, which isn't a lot.
However, Harry is the single person in history who was bestowed the terrible death curse and lived. And not only that, the curse also bounced back to the Dark Lord and disabled him for over a decade, whereas before Voldemort had been more or less unchallenged and unstoppable(*).
This is what distinguishes an entirely unimpressive boy with no special abilities from a total loser and makes him the protagonist that stands out of the crowd.
It is, as stated by Ollivander, remarkable that Harry is chosen by the very wand whose brother gave him the scar. Which leads to another plot device which again, for the third time allows a boy without much skill or talent to escape death, against all odds, when battling the most powerful evil wizard in history (who masters the death curse against which no defense exists).
All of this wouldn't work if the killing curse was only just another way of killing a person. It must by definition be the most terrible curse, in order to stand out.
Otherwise, if only some other, unspectacular curse had been cast on Harry as a baby, one would have to dismiss his survival as "accidential" or "being lucky" (since he obviously didn't survive due to his outstanding wizardry skills, maybe Voldemort just didn't aim properly). In other words, there would be nothing special about Harry.
The killing curse is not "particularly terrible" otherwise, except for the fact that you die. It is described in the books as a green light, and then you are dead. There is no mention of pain or a particular cruelty whatsoever. The depiction of what happens to the victims (such as e.g. Cedric Diggory in book 4) is very much suited for an "all audiences" rating. The lifeless body simply drops dead. Neither does the victim cry in agony or end up curled up in convulsions. In the later books, people who have actually experienced the death curse testify that there is not much to be afraid of, it is not painful and it is over pretty quick. From the reappearance of several characters as "ghosts" or "souls", one can conclude that the curse is not particularly devious insofar as it e.g. destroys (or robs) the victim's soul either.
Arguably, the killing curse might appear "more terrible" than other spells from an ethical point of view. Other spells may certainly cause a much worse death experience, such as for example a fire spell. Burning alive is without doubt a lot worse than seeing a green light and dropping dead. However, a fire spell, like most magic, may have other legitimate uses whereas the killing curse serves only a single purpose, snuffing out life. There is, however, no mention of any such thing in the books. This would be purely a reader's conclusion.
The death curse's main purpose remains being a plot device for drama and for giving the main character a justification in the story.
If the death curse was a real element of the story's universe, one would have to wonder why Voldemort and his Death Eaters are not killing everyone who gets in their way (including for example Dumbledore) on the first occasion. After all, there is no defense against it. Why would you use a spell that can be countered, if you have a spell that just kills anyone? How could a Death Eater ever (reasonably) get captured by an auror? Why would you fear Dumbledore if you can just snuff him out in an instant?
It took Voldemort a complete six books to figure out that he could send one of his followers, even a boy, to kill Dumbledore with the death curse. Malfoy didn't have the guts to do it, so this finally remains unknown, but apparently it no problem for a mildly gifted adolescent (from a technical point of view) to kill the most powerful wizard alive.
Yet, throughout the by far dominant part of the story, the death curse is only used for killing sideshow characters like janitors and school boys, merely to keep up the reader's memory that such a thing exists, and the fear of the unknown. The Dark Lord may strike at any time, and only one person is able to resist him.
This is however inconsistent with the later books which were in my opinion written by ghostwriters that didn't know (or care) much about the original stories.