PC or NPC is not a property of a character in the fiction.
Correct. We agree on this, at least.
It's a description of who among the game participants owns/controls a character.
Indeed. The question then becomes, why should who controls a character make any appreciable difference to that character in terms of its game mechanics? (I take as a given that the mechanics are the framework on which the character is made and played, in other words that the character is played more or less to reflect its stats etc.)
So the real question is Why do the players get to control the implacable warriors, while the GM is stuck controlling the sea of mediocrities? And the answer is because - as per the blurb on the back of the PHB, THE WORLD NEEDS HEROES, a premise of the game is that the players control the protagonists.
I ignore blurbs like that.

There's no heroic expectations here: they can be villains if they like, or explorers, or whatever.
The players control active people in the setting. Whether those people are themselves protagonists or are instead reacting to the protagonism of others is going to, one hopes, be changing all the time as the campaign goes on.
(side note: the bolded is the best line I've seen today!)
The character operates under the same "rules" as everyone else: they draw their sword (or whatever) and engage their foes in melee. They just happen to be implacable, relentless, remorseless, however exactly you wish to characterise it. Like Conan, or Eomer, or Aragorn, or Lancelot.
All those examples are of highly-experienced and well-trained warriors who would in theory map to very-high-level D&D characters. Joe the 2nd-level Fighter isn't yet anywhere near that good and IMO isn't all that much different than an experienced soldier in an army, other than having a desire to operate more independently.
To me, it seems like complaining that (say) Cathy Freeman (Olympic 400m, Sydney 2000) or Gary Kasparov (champion chess player) or Rasputin (notoriously hard to kill) doesn't operate under the same rules as everyone else.
And yet in almost all instances they do (or did, in Rasputin's case). They are all human, they all need to eat and sleep like the rest of us, they all have to abide by the same (or very similar) societal norms and laws, and absent prior knowledge one could pass any of them in the street and not notice a thing.
Those people are each far more similar to you or I than is a typical adventurer to a commoner in a D&D setting.