Tolkien said LotR was "a fundamentally Catholic work". And by that I see
not that it reproduced the doctrines of the Roman Catholic church, for that the reader has to turn to other writers, but it reflects certain parts of that faith - perhaps the Valar/Maiar could be seen as versions of saints or angels in a fundamentally ordered, yet very different universe.
LotR does not deny God, so it is not an atheistic or agnostic work. But is it a secular work? Whether Tolkien wanted to exclude symbols of religion or not, they are still to be found in his work. I think this is due in no small part to the fact that many of those symbols, getting away from the obvious ones such as the crucifix or crescent moon, are ingrained within us; archetypes might be the correct term
. As just one example we have the semi-Trickster figures recently discussed. Or a link can be drawn between Galadriel and Brigid, with the significance of water common to both. I think religions take powerful or familiar symbols from the world about us (or from worlds that were once about us) and make use of them; it is inevitable that Tolkien would have unwittingly/unconsciously used some the less obvious symbols.
I can see exactly why Tolkien would not want to include the more overt symbols of religion in his work. He was creating his own world and those symbols would not belong there as this world would need its own significant symbols, such as the Star of Earendil. Any overt images would make his work not fantasy but something different.