Homer is a particularly American archetype, just as Doug and Bob McKenzie could only be a Canadian one. I don't see anything similar to the Homer in Greek mythology, but he does exist in the Judeo-Christian tradition, as a modern-day Adam (which means "Man").
Homer is good-natured, but stupid, ignorant, loud and obnoxious. He means well, but fouls up everything he does. He blames his problems on others, but every once in a while he has moments of self-awareness in which he realizes he is his worst enemy. His worst sin appears to be that of Intellectual Sloth (eyes rolled up toward cranium: "Brain, I know that you and I don't get along very well..."). If you were to take a million Americans, boil them down, then take the concentrate and morph it into a cartoon character, you'd have Homer. That is why he is such a uniquely American archetype. I think this is why he is so popular: the Homer in all of us resonates with the Homer onscreen.
Homer has his moments, but overall he is not likeable. He's not evil, just stupid (which reminds me of the comment by Spider Robinson and Robert A. Heinlein: "Never attribute to malice what can be explained by stupidity"). Homer is funny in fantasy, but like a lot of things which are funny while unreal, he'd be a horror in real life. Laughing at him allows us to defuse the fact he is a catastrophic symbol of ourselves.
Homer fits the archetype of Adam because he unconsciously blames his problems on other people. This is what Adam did when he blamed Eve for his transgressions. Homer, however, in his brief periods of self-awareness (always heralded by the now-classic "Doh!"), realizes he is at fault. He goes from an unconscious scapegoating to a conscious acceptance of self-responsibility, however transient.
This moving from unconscious scapegoating to conscious acceptance of self-responsibility is why I believe the story of the Garden of Eden to be the most powerful of all. The psychiatrist M. Scott Peck quite correctly called scapegoating "the genesis of human evil." And Homer, proving him right, has an almost irresistible impulse to blame his problems on everyone but himself.
I've often wondered if Adam had only gone, "Doh!" and accepted responsibility instead of blaming Eve (and had Eve not blamed the Serpent), then God might have said, "Well, okay, you can stay." In some versions of the story, Adam and Eve's scapegoating, and their denial of their responsibility, is what gets them kicked out of the Garden.
Since each of us has an Inner Homer, we are imperfect. In traditional religious terms, we are "fallen." Since everyone in The Simpsons is imperfect, the program can be considered in some ways to be a "true" conservative show, since true conservatives know that no one is perfect. This, of course, includes everyone in the government, which is why Chief Wiggins made the classic comment: "I didn't say the government couldn't hurt you. I said it couldn't protect you."
Since the program is very tolerant of everyone (and their foibles), it's also quite libertarian. Libertarian conservative, you could say. Except for Lisa, who in her youth and foolishness is a liberal. She's probably an example of that old saying, "He who is not a liberal at 20 has no heart, and if not a conservative at 40, no head."
Did the creators of Homer know what they were doing when they projected an aspect of the national psyche into this character? I have no idea. Like many artists, they tend to be, as Ezra Pound commented, "the antenna of the race." They can often see how things are (and are going to be) more clearly than most people, even if they can't explain how they know it.
Since Homer is a projection of our selves, he is Mass Man. The mean average IQ is 100, which is about what Homer's appears to be. And the more Americans you gather into a group, the closer the IQ drops to 100. This is one of the reasons I think the Essence of One Million Americans is Homer. Whenever you see some fat, yelling, shirtless, body-painted drunk at a sports event, you're seeing the Doh! made Flesh.
One problem for America is that this concentrated Homer archetype is what votes. Hence, Homer America. This is why democracy doesn't work, and why we get the politicians we do. Homer is definitely not part of Albert Jay Nock's aristocratic and educated Remnant. For that matter, neither are politicians.
Unfortunately, the bigger the group of politicians, the more they become Homerized, just like the average citizens. This is a great example of why government should be as small as possible. The bigger it gets, the more Homeresque it gets.
The rest of the world, sadly, tends to see the U.S. as Homer: big, good-natured, somewhat stupid and ignorant, loud and obnoxious, meaning well, but screwing up everything it gets involved in. And every once in a while, the U.S. has its moments of lucidity in which it realizes what it's doing.
The U.S. and the average American weren't like this in the recent past. TV fathers like Homer didn't exist until the '90s. In the '50s and '60s they were Ward Cleaver and Andy Taylor. And the US was a lot more respected by foreigners than it is now.
If The Simpsons had been on the air in the '50s and '60s, would it have survived? I doubt it. It's too insulting to most people. The closest to The Simpsons was The Flintstones, and Fred wasn't a couch-potato quasi-drunk like Homer.
As funny as The Simpsons is, it certainly is quite a drop for the U.S. and fathers to go from Andy Taylor to Homer.
What Homer may show is that the greatest sin of Americans is Intellectual Sloth. This makes a great deal of sense, they are consisting stumbling into all kinds of world catastrophes. A lot of Americans are afflicted with Intellectual Sloth of Homerian proportions.
What has caused this change? I'm not exactly sure, but I suspect the precipitous decline in education has something to do with it. This is why it is significant that Homer is ignorant and his worst sin is that of Intellectual Sloth.
Homer shows us at least four things: people are imperfect; they no longer understand (if they ever did) the death and destruction that unconscious scapegoating can lead to; that democracy is an awful form of government; and that the public schools should be closed down. The idea of Homer's hometown of Springfield growing like the Blob until it encompasses the U.S. is not a pleasant thought, no more than the idea of public schooling engulfing the country is pleasant.
Homer can be considered a warning to America: this is what we have become. And while he's been defanged by being turned into a cartoon, when he moves into reality, catastrophes happen. I shudder to think of the US being run by Homer.
Oh, no, wait...