Tolkien often talks of "discovering" vrs. "inventing" in his writing.  I myself find the same to be true in my own art form.  Essentially, what I think this adds up to is: When one is creating and expressing something (be it: a story, a feeling, an expirience, an idea) their personal beliefs, history, and accumulated knowledge is unstoppably wound up in the fabric of its being.  This is most likely why many artists are so hyper-sensative to a negative comment.

    I agree that LOTR is fundamentally a Catholic story, but it is also fundamentally a "Tolkien" story.  As he said himself, "if it is childish in parts, that is because of my own childishness." 

     Though I'd argue that the story could not exist without Tolkien's faith- I'd also argue that one cannot anazlye whether it is a blessing or a detriment because it is what it is.  It is the culmination of WHO Tolkien is-  how he sees the world, how he has lived, what he feels...etc.  One cannot simply remove a thread of that fabric without unraveling the entire history and story.  Gandalf did say, "he that breaks a thing..." 
     As a Catholic reader, I am very interested in seeing concepts I believe in mirrored in his story.  As for non-Christians, I'm sure they can connect with the pure story with ease.  I think Tolkien's comments on 'varied applicability' are extremely wise- and this allows all readers with imagination to grab onto the story and take from it what they wish.  Isn't that what we do with almost anything/anyone/any expirience?

"Will you look into the Mirror?"