You forgot to note that Tom Shippey finds that LOTR is not religious.

"Frodo offers no promise of soul-salvation (though he has saved Middle-earth from a great danger), he releases no prisoners from Hell (though he does from Sauron's dungeons), he does not rise from the dead (though Sam for a moment, and entirely understandably, thinks something like that might have happened). Frodo in other words has no supernatural dimension at all."

He adds:

"The myths of Middle-earth furthermore determinedly reject any sense of ultimate salvation."

Indeed, by putting a non-religious event in the "mythic space" of the Christian calendar, Tolkien replaces Christianity with non-religious story.