I watched that video a few times and thought it was pretty funny, but more than that I was just enormously confused (and intrigued!) about the context. What the hell is going? Why is she seducing him with a parodical book number? What are regionals anyway?
So. Every song in this episode is, in itself, fundamentally false and deceptive. The plan that Abed and Troy hatch to undermine Christmas is explicitly hypocritical, since Troy plans to go on celebrating Christmas forever ("I might have to dedicate my life to Christmas / And act just like I love it til the day I die"). Baby Boomer Santa, as Annie points out, relies on revisionist nostalgia to flatter Pierce's "generation's well-documented historical vanity." Annie's song to Jeff is, as Jeff perceives, a "bit." The song that infects Shirley is a cynically saccharine concoction designed to push her buttons, lampshaded in the line about baking but also present in the political lyrics, which sound so out-of-place coming from children. The original song about glee is a lie Mr. Rad tells Abed, which Abed then repeats himself, and this lie is the central one examined and exploded throughout the episode: the idea that "everything's cooler when cameras are spinning," or said more plainly, that things are better when you disguise the truth, when you "try to make things brighter."
None of these songs are "heart-songs" that come from deeply felt emotions. Sometimes they're a lie the singer tells the listener, such as Annie telling Jeff she's sexually available, or Troy and Abed telling Pierce his generation created everything worthwhile. Sometimes they're a lie the singer tells him- or herself, such as Troy convincing himself celebrating Christmas is consistent with his religion, or Abed telling himself that glee will bring his friends "to a healthier place." The fundamental act of lying is present in each song.Abed is aware that they're all being deceptive throughout the episode. Lines like "Glee / Is what I'll spread to my friends, like a virus..." and "I might have a loophole" suggest that, unlike the others, he remains aware that what they're doing is fundamentally dishonest. But he's willing to be dishonest in order for everyone to be together and (at least apparently) happy at Christmas. At the glee club performance at the end of the episode, however, Mr. Rad makes him realize that once you start lying it can be hard to stop. Rad's obsession with regionals mirrors Abed's obsession with Christmas: they're both milestones that have become personally important, but are actually intrinsically
Fortunately, Abed has the perfect tool to unravel all of the accumulated falseness: ugly truth. As I mentioned, I don't know much about the characters in terms of their arcs and underlying identities, but I do gather that Brita is an iconoclast, and I wonder if Rad's abhorrence for Brita has something to do with that, with a fundamental fear that she'll destroy the icons that make up his world. In any event, Abed tells Brita to sing her from her heart, and she uncynically does. Obviously the words of Brita's song don't express any deeper truth, but her singing, sans glamor and accompaniment, exposes the emptiness of the whole gleeful enterprise. Even more, it brings forth a much uglier truth from Rad himself, revealing that living so disingenuously is not only empty, but actually dangerous. If your life becomes devoted to lies, you're forced to become more and more callous towards real people in order to maintain them.
Incidentally, I found the coda scene of this episode, with the cast singing "The First Noel" for Abed, a little disappointing on first viewing. It just didn't pack the emotional wallop I was looking for. But I think that's the point. The episode can't end with swelling music and a group hug, because that's not where the group is. But it can end with a sweet gesture and a choice to be together, however reluctantly made. The choice of the carol is a very nice touch: "The First Noel," being about the first appearance of the angels to the shepards, emphasizes the arrival of salvation in a time of great darkness. Using this song makes the point that music can be honest without being hopeless, that our communication with each other can be both truthful and kind. This isn't one of my favorite carols, but the more I think about it the more I like its use here. Some other good choices might have been "We Three Kings," the first verse of which is about a journey to get to Christ, or the wonderfully ambiguous beginning of "O Little Town Of Bethlehem":
O little town of Bethlehem, how still we see thee lie