Of course the story has Christian elements, but it also has non-Christian elements. Tolkien was a devout Catholic, and this influenced his thought, and was clearly an element of what was going on, but it really was only one element of what went into LOTR and the rest of his opus. His professional obsession with languages and old Northern European myth also played a substantial role in his fiction.
Tolkien himself was mixed in his statements about the religious element of his writing. He did state what you wrote above, but later he said this:
“It is not about' anything but itself. Certainly it has no allegorical intentions, general, particular, or topical, moral, religious, or political. The only criticism that annoyed me was on that it contained no religion' (and no Women', but that does not matter, and is not true anyway). It is a monotheistic world of natural theology'. The odd fact that there are no churches, temples, or religious rits and ceremonies, is simply part of the historical climate depicted. It will be sufficiently explained, if (as now seems likely) the Silmarillion and other legends of the First and Second Ages are published. I am in any case myself a Christian; but the Third Age' was not a Christian world.” Tolkien, Letter 165.
He also seemed skeptical about critics who would read too much into his work in the religious sense, I think, when he said this:
“..I object to the contemporary trend in criticism, with its excessive interest in the details of the lives of authors and artists. They only distract attention from an author's works (if the works are in fact worthy of attention). and end, as one now often sees, in becoming the main interest. But only one's guardian Angel, or indeed God Himself, could unravel the real relationship between personal facts and an author's works. Not the author himself (though he knows more than any investigator), and certainly not so-called psychologists'. ...I was born in 1892 and lived for my early years in the Shire' in a pre-mechanical age. Or more important, I am a Christian (which can be deduced from my stories), and in fact a Roman Catholic. The latter fact' perhpas cannot be deduced; thou one critic (by letter) asserted that the invocations of Elbereth, and the character of Galadriel as directly described (or through the words of Gimli and Sam) were clearly related to Catholic devotion to Mary. Another saw in waybread (lembas)=viaticum and the reference to its feeding the will (vol. III, p. 213) and being more potent when fasting, a derivation from the Eucharist. (That is: far greater things may colour the mind in dealing with the lesser things of a fair-story.)” Letter 213.
Tolkien, I think, basically intended LOTR to have some Christian themes, together with other themes that are common to heroic literature, particularly Norse mythology (from whence come Elves, Dwarves and even the term “Middle Earth”, which is simply the English literal translation of the Norse “Midgard”, the place in the Norse cosmology where mortals lived). It is a mixed work.