The word "human" is used a handful of times in the Legendarium, but always in ways where it's ambiguous whether or not Tolkien is acting as a translator. For example, in The Two Towers (emphasis mine):
So they came slowly to the white bridge. Here the road, gleaming faintly, passed over the stream in the midst of the valley, and went on, winding deviously up towards the city's gate: a black mouth opening in the outer circle of the northward walls. Wide flats lay on either bank, shadowy meads filled with pale white flowers. Luminous these were too, beautiful and yet horrible of shape, like the demented forms in an uneasy dream; and they gave forth a faint sickening charnel-smell; an odour of rottenness filled the air. From mead to mead the bridge sprang. Figures stood there at its head, carven with cunning in forms human and bestial, but all corrupt and loathsome.
The Two Towers Book IV Chapter 8: "The Stairs of Cirith Ungol"
Or Unfinished Tales:
Likewise within the lands the birds of Númenor were beyond count, from the kirinki that were no bigger than wrens, but all scarlet, with piping voices on the edge of human hearing, to the great eagles that were held sacred to Manwë, and never afflicted, until the days of evil and the hatred of the Valar began.
Unfinished Tales Part 2: "The Second Age" Chapter 1: "Description of the Island of Númenor"
However, there is at least one occasion in the Lord of the Rings where Tolkien draws explicit parallels between his "Men" and our own humans: the prologue.
The prologue is different from the rest of the story, because it's written in-universe in Tolkien's own voice. In one section, Tolkien specifically groups himself, and, by extension, us, with the Men of his story (emphasis mine):
Hobbits are an unobtrusive but very ancient people, more numerous formerly than they are today; for they love peace and quiet and good tilled earth: a well-ordered and well-farmed countryside was their favourite haunt. They do not and did not understand or like machines more complicated than a forge-bellows, a water-mill, or a hand-loom, though they were skilful with tools. Even in ancient days they were, as a rule, shy of 'the Big Folk', as they call us, and now they avoid us with dismay and are becoming hard to find. They are quick of hearing and sharp-eyed, and though they are inclined to be fat and do not hurry unnecessarily, they are nonetheless nimble and deft in their movements. They possessed from the first the art of disappearing swiftly and silently, when large folk whom they do not wish to meet come blundering by; and this an they have developed until to Men it may seem magical. But Hobbits have never, in fact, studied magic of any kind, and their elusiveness is due solely to a professional skill that heredity and practice, and a close friendship with the earth, have rendered inimitable by bigger and clumsier races.
It is plain indeed that in spite of later estrangement Hobbits are relatives of ours: far nearer to us than Elves, or even than Dwarves. Of old they spoke the languages of Men, after their own fashion, and liked and disliked much the same things as Men did. But what exactly our relationship is can no longer be discovered. The beginning of Hobbits lies far back in the Elder Days that are now lost and forgotten.
Fellowship of the Ring Prologue Chapter 1: "Concerning Hobbits"
Now, regarding the theory posed in the question, this explanation does open up the possibility that the narrator is also one of Tolkien's "Men", rather than a human of our own species. However, in-universe the narrator is most assuredly J.R.R. Tolkien; this is discussed more fully in this question (disclaimer: answered by me).