Given the lyrics Cohen sang himself, the original song was a downbeat anti-gospel that had nothing to do with God, and more to do with sex and relationships. Jeff Beckley cleaned it up a bit, but it's been the modern singers and players, especially Pentatonix and Lindsey Sterling, who've given it some pro-God depth (despite Cohen).
I think this is because Cohen wanted to use the depths of dispare to show that calling "let it be so, God" or "Glory to God" at the bottom is cynical. Instead, it is human, and what God wants. When we've failed or we're at the bottom, God wants us calling to Him. If we're getting punished rightly in our sins, or if we're in a terrible position through no fault of our own but because God is with us, the only appropriate thing to say is "Your will, God". Saying "let it be so, God", one meaning of the original Hebrew word for Hallelujah, is what we have to do.
While we can interpret it as pseudo-Gospel song, it's not a Christmas song.
The Pentatonix version:
Lindsey Sterling's version:
Merry Christmas and Happy New Year!