If the author says it’s not an allegory, and it doesn’t work as an allegory, maybe their identifying as Christian isn’t enough to make it an allegory?
As a non-believer, I quite enjoy the lack of anything overtly religious in his works.
That comes through in a fascinating sense of alien-ness. The books feel both realistic and unfamiliar because he's capturing a sense from the past as a very foreign country. That's a sense available to you as a non-believer because it's not gussied up in the specific trappings of any belief system you've been exposed to (and probably put off by).
The highest virtue in The Lord of the Rings is a very Christian kind of hope -- hope for the next world rather than this one. And the key virtue of humans, rather than Elves, is that they get to go there, while the immortals are trapped here. It's easy to miss all that in The Lord of the Rings because it does ultimately depend on hope in this world -- they win and evil loses.
But look at the sadness of the Elves, who know that they will ultimately lose and the world will decline. Think of that from the point of view of somebody living among Roman ruins that they cannot repair. And the only hope comes from a far-off entity promising a better afterlife.
It's not propaganda, but it's a very real demonstration of the virtues of his outlook.
JK Rowling is thought to go back and retrospectively reframe her book, a framing which she has trouble gathering consensus for despite being the author.
The Letters of J.R.R. Tolkien #142