I don’t know if people still complain about Christmas songs being played early; like most things about “people” I suppose it depends on who one talks to. Anyway, while I’m sympathetic to the idea of “keep the waiting in advent,” it has occurred to me that there is a reason that recently traditional secular Christmas songs are song before Christmas and not after: if you look at them, they are really advent songs. Secular advent songs, of course, but advent songs. (I’m taking the list of Christmas songs from XKCD’s list which I discussed earlier.)
Have Yourself a Merry Little Christmas and Have a Holly Jolly Christmas both have titles (and main lyrics) in the future tense. Santa Claus is Coming To Town is technically in the present progressive tense, but all of the lyrics are anticipatory—primarily warning about present behavior in light of future rewards. Chestnuts Roasting on an Open Fire is set on Christmas Eve, but that is still, technically speaking, during Advent (unless you’re measuring days from sundown to sundown, in which case I think that the present-tense of the song would have to be taken as anticipatory).
I’ll Be Home for Christmas, though I rarely here it played or sung, is another one clearly set in the future tense and thus an advent song. I’m Dreaming of a White Christmas would most naturally be taken to be about anticipation though it could, technically, be set on Christmas. That is, until you get to the later lyrics where he dreams of a white Christmas with every Christmas card he writes. It would be absurd to suppose the song is about somebody who sends out Christmas cards after Christmas, since their purpose is to wish someone a happy Christmas.
Rocking Around the Christmas Tree is harder to place, temporally. Its subject is a Christmas party, which I’m used to being held prior to Christmas but in 1958 when it was released it might have been the custom to have Christmas parties on Christmas day itself, though I am inclined to doubt it.
Blue Christmas (which, again, I never hear anyone sing and don’t hear played) clearly talks about Christmas in the future tense in the lyrics (“I’ll have a blue Christmas without you”).
Silver Bells could be set on Christmas or even after it. That said, it’s about Christmas decorations and such which are generally put up before Christmas, so the smart money is on it being an anticipation of Christmas.
It’s Beginning to Look a Lot Like Christmas is another one whose very title shows it to be set before Christmas.
It’s the Most Wonderful Time of the Year is, like Silver Bells, not explicit, but it seems to be about the (secular) season of preparation for Christmas, placing it before Christmas.
The other songs on the XKCD list (with one exception) aren’t about Christmas at all, or at least not a present Christmas. Winter Wonderland, Let It Snow, Jingle Bell Rock, and Sleigh Ride are all just about winter. (So, for that matter, is Baby It’s Cold Outside, which is increasingly be played as if it’s a Christmas song.) Rudolph the Red Nosed Reindeer is primarily about the time before Christmas, and culminates in Rudolph’s triumph on Christmas Eve, but al of this was in the distant past. Frosty the Snowman is about a magical snowman and has nothing whatever to do with Christmas. (Admittedly, the animated movie Frosty the Snowman is set on Christmas Eve, but that’s still anticipating Christmas.)
The only real exception on the entire list is Little Drummer Boy, which is actually set after Christmas. It seems to be based on the visitation of the Magi, which is traditionally celebrated on Epiphany, which for many years in the western Church has been celebrated in January. Since the song doesn’t reference anything that sets its date, it could be anywhere from the day of Christ’s birth (e.g. when the angels gave the good news to the shepherds) to months after the Magi visited. I suspect that no one pays attention to the lyrics of this song, though, since approximately 20% of them are “pa”, 20% are “rum” and 45% are “pum”.
So, all things considered, I think we have some of the reason why these songs are all played before Christmas, rather than after it—they are, in fact, (secular) advent songs. As Chesterton often noted, the common man often has his heart in the right place, even when it’s there for the wrong reasons in his head.