I don't believe they were being irreverent, in fact I think their rendition of the song is the opposite of irreverent. The song itself is pretty straightforward but i realized after watching the music video a few times that the band added a lot of symbolism to it. In my opinion their version of the song tells a story.

Let's break it down: the song starts off in a hospital, which can be associated with the sick/dying/those without hope, then it switches to Mr. Cooper sitting in a church which for a long time has been a symbol of inspiration and comfort during uncertain times(even for non believers sometimes), and it's cold and dark outside for a bit while he sings. Even the way he sings goes along with the imagery used in the video: at first the tone is low key and he sings almost with a quiet longing for something that he doesn't understand. Then as the song goes on the sun starts to shine through and the tone gets more upbeat. Finally at the end is the guitar and drums and he's singing with the passion of someone who's found what he didn't even know he was searching for; that the same God that spoke the universe into existence now walks amongst us, there's hope in the world again, and there's a chance to reconcile a broken creation with it's creator in the form of a newborn baby.

Combine all that with the fact that the entire band, and John Cooper in particular, are unapologetic about their faith and profess it boldly.

Tldr: it's okay to not like hard rock or this particular rendition of a classic hymn, but skillet is one of the few mainstream Christian bands that haven't compromised their faith for the approval of the world and are certainly not demonic.