People are actually somewhat hard to kill.
In some cases, there's no logical reason the player wouldn't be able to kill the enemy easily enough. But actual living beings have adapted for survival, and it's remarkable just how much something might survive, even if it's only for a while.
With D&D healing magic, survival is even more likely if the enemy is in the vicinity of their friends.
(Warning: these links might be considered graphic, although I didn't see anything too serious in them. Except the one with the bolded warning; it's decently graphic.)
- Heart shots. Contrary to Hollywood depictions, shooting someone in the heart doesn't kill them; bleeding out does.
1.1. After a heart shot a deer can continue running for up to two minutes if they haven't been actively using up their anaerobic energy supplies with physical exertion, though they generally pass out from loss of blood pressure after 10 to 15 seconds (or instantly depending on the hit). That's 2 to 10 game rounds of potential fighting.
1.2. Arrows and knives aren't as lethal as bullets because they don't have the velocity, and they help plug the hole if they aren't removed. Numerous people have survived crossbow bolts to the heart, such as this guy who survived a suicide attempt even though a bolt penetrated his heart, and this guy who not only survived the first shot through his heart, but loaded a second bolt, fired again, and was still saved.
- Head shots. Even head shots aren't always fatal.
2.1. This 11-year-old boy survived an arrow that went through his eye and hit the back of his skull.
2.2. This U.S. Army soldier was stabbed through the side of the head with a knife, and not only survived, but returned to active military service two years later.
- Slitting a throat. The jugular and carotid are pretty well protected from slashing by the cartilage around the windpipe, and it's relatively easy to miss a stabbing attack on the first attempt if you don't know exactly where to aim.
3.1. A man in New Zealand survived a guy slashing his throat from behind, and was able to walk to a nearby dairy several minutes later.
3.2. A medical report (this one is more graphic than the others) tells of three patients who survived being stabbed or sliced in the neck, one of which (the third) was a deliberate suicide attempt. The second case (the barber had a seizure while shaving the client, slicing his neck; yikes!) cut into the jugular, but the guy still survived. The introduction of the report notes that "penetrating neck trauma" was fatal 11% of the time in World War I, 7% in World War II, and 3-6% in civilians today. Obviously, that includes accidents and so forth, but even gunshot wounds are potentially survivable.
But these are trained killers, right?
That really depends on your player's level, class, etc. It stands to reason that a level 10 fighter probably knows how to stab someone in the neck so they're pretty much guaranteed to bleed out. But, like others have mentioned, if the guy is actively trying to avoid dying, he's probably reasonably good at knowing how to move out of the way (if he's high level), or a standard attack will kill him anyways (if he's low level).
On the other hand, a level 1 wizard might not even have the strength to get his dagger through the bone or cartilage consistently, even when the opponent is in a long-term coma. So some kind of roll would definitely be appropriate regardless of the enemy's status.
Additionally, D&D is full of different creatures who might have very different anatomy than a human, making it less likely even a skilled fighter would be certain of a kill shot.
Finally, the NPCs are trained killers too. One part of not dying involves protecting themselves from things like obnoxious adventurers trying to one-shot you. Since NPCs will likely wear pesky things like armor, your player not only has to aim in the right spot, but penetrate a quarter inch of chain links (or whatever). And with magic, there could be any number of effects that would prevent an instant-kill, particularly on bosses and rich people.
House rule ideas.
I don't know much about 5e, so don't take this part too seriously.
In keeping with the above, you might:
1. Limit auto-kills to higher-level characters (perhaps tied to feats involving specialized killing, like Martial Adept or Sharpshooter).
2. Require an Intelligence or Perception check to see if the player knows where to strike (maybe with a bonus for player level, and a penalty for non-human creatures).
3. Require a successful attack roll even if you don't require a damage roll (attack with advantage if the target is truly immobilized, normally if they can squirm, and with disadvantage for alien creatures or player classes not likely to be trained in anatomy). A failed attack roll might represent a normal hit in this case, requiring a standard damage roll, waking up enemies who aren't comatose, and allowing them to scream bloody murder to alert everyone around.
4. Auto-kills shouldn't be possible if there's any way an enemy healer could get to them in time to stop bleedout and brain damage. Nor when killing the enemy would logically initiate combat (give the enemy's friends a chance to save them).
I think there are cases where an auto-kill just makes sense. If the guy is trapped in a cremation furnace and you turn it on, game over. If the guy is completely comatose, the difference between combat rules and auto-kill is just how much time you waste getting to the inevitable. Etc.
But people can survive pretty crazy amounts of damage, even direct shots through their heads, hearts, lungs, and necks. Further, armor goes a long way towards preventing death if the players haven't removed it.
And D&D is full of magic and alien wonders that can be used to make enemies even more resilient than real-world humans.
So at the end of the day, requiring players to make attack rolls against helpless targets isn't just part of the rules. It can actually make sense.