Seeing as this is The Onion, the anvilicious message is basically lampshading the trope, by mocking serious news reports on the problems. It's in dire need of pointing out, remembering the Bangladesh incident. It should also be noted that there about 50 Klingon speakers (at the most) and about 170,000 Navajo speakers, thanks to revitalization efforts.
The Onion is often anvilicious non-ironically, as the writers' very liberal bias is worn like a badge. One recent article was simply about people in the future scolding people in the present for not legalizing gay marriage (also, in the future, every single person in the entire world is pro-choice).
The Big Finish Doctor Who audio The Last is ridiculously anvilicious in its anti-war message. It takes place in a post-apocalyptic wasteland, with statements like "Money that should have went to space exploration went to develop more weapons" and "She should have known dropping bombs is wrong, that war is wrong".
During World War II, many radio shows devoted some time to explaining and/or promoting various wartime programs (rationing, War Bond drives, etc). Most of them were less than subtle about it. Such as this infamous poster◊.
The truly Anvilicious narrator in Blood Brothers not only shows up to highlight every moment of foreshadowing in the musical, but also appears at the end to let any terminally inattentive audience members know what the message was.
The Toxic Avenger, based on the campy movie of the same name, could be a hilarious hour of nothing but New Jersey jokes, which it is in some places, but eventually it gets bogged down by its need to hammer "Pollution BAD!" into the audience. Emphasis on hammer.
Brad Jones reviewed Rock: Its Your Decision, a little-seen TV movie produced by fundamentalist christians about how rock is evil and how it'll cause you to go to Hell. It doesn't merely lob anvils, it ties them to cruise missiles and fires those at you. After the main character of the movie talks about how people at rock concerts didn't just sit quietly and listen, they got up and danced, like the music was controlling them, Jones jokes that the movie's Aesop is that emotions are bad and if you dance to music, laugh at comedy acts, or cry at funerals, you'll go to Hell.
Mr. T, in every incarnation, is anvilicious to the point of becoming a running joke, thanks in no small part to the ironic humor of a large violent macho man screaming at you things a meek female kindergarten teacher would normally tell you. This resulted in a favorite satirically jumbled line from SNL: "If you believe in yourself, eat all your school, stay in milk, drink your teeth, don't do sleep, and get eight hours of drugs, you can get work!"
The weird thing about Mr. T is that he means all of it; the man who made a career out of screaming and hitting things is a big ol' Momma's Boy who loves kids and wants to use his image to help them. In the '80s he performed in a video called "Mr. T's Be Somebody or Be Somebody's Fool," a series of vignettes encouraging self-confidence and good decision making that is so anvilicious the videotape would make a pothole if you dropped it in the street. Unlike most celebrities doing pre-teen educational/self-help/anti-drug videos in the '80s, Mr. T was not ordered to do this as the community service portion of a drug sentencing; he co-wrote and produced the thing of his own volition. That's because Mr. T don't give ya no jibberjabber. He tells it like it is.
During the 2011 Superbowl, Will.I.Am of the Black Eyed Peas took a moment during the halftime show to tell Obama to reform the school systems.
Practically every child has had to attend an assembly at their school about how drugs and alcohol will ruin their life. In some health classes, its not uncommon to have a "speaker" come and show the students pictures of an STD infection.
Another common problem is that sex educational videos buy into the All Abusers Are Male trope hard. Every time someone is being pressurized into pre-marital or underage sex they are Always Female. So the lessons become less "be safe, sane, and consensual," and more: "GIRLS! Defend your virginity against the be-penised hordes!" Which of course leads to nobody taking them seriously at all.
A particularly hilarious one the Scottish "well-meaning but comically futile campaign to make children take adult things seriously" educational curriculum involves one lesson showing teenagers how being immature about sex is bad, mmkay, which begins with splitting the class into groups and asking them to write down all the different terms for penises, sex, vaginas, homosexuality, and contraceptives, without fear of punishment. These are then read out by the teacher. The intention is to engage the class and make them realize how they buy into sexist or immature attitudes. Unfortunately, this is Scottish teenagers. The end result is usually the class roaring with laughter, whilst a red-faced (and frequently deeply religious) teacher goes "boaby...fanny...walloper...Glasgow typewriter...batting from the wrong end of the crease...bellend...tuna sandwich", leading to the children thinking less "sexist and immature swearwords are bad" and more "god, being immature is hilarious."