I'm fine with a well-written idiot and one that expresses abilities to, I dunno, survive. Some characters, such a Patrick from Spongebob (in recent seasons), have degraded into what they are now. Patrick, while never bright, was never as stupid as he is currently. They also might be so far gone that they abandon their sense of judgment, morals, or friendship (*cough*Cosmo*cough*)
I also wouldn't mind some common sense. I think the best idiotic characters are ones that might misinterpret or get involved in some crazy stunts. However, they are goodhearted at their core and intelligent to some degree. So, I don't know if they necessarily qualify as an idiot.
I suppose they are added because a portion of the audience might love them or because they, as mentioned, are easier to write. Some mistake their behavior for a personality, when they should really try fleshing a character out. Being "the idiot of the show" doesn't mean that they never understand anything or aren't capable of learning, nor does it mean they lack any other emotions/traits/faults. Another downside is that the characters always seems to have edge over the rest. When they do something wrong, they're 'so humorous' that they don't show regret or remorse and everyone quickly moves on. There are exceptions of course, but I'd rather see a character struggling with him (or her) self, knowing they may not be as intelligent as the rest. Rarely do I see shows that tap into those emotions unless the character in question is already intelligent and has a rival.
In general, I define an idiot character in different ways; one that should not have been able to survive because of their outright stupidity, ones that aren't that bright, granted, but not to the extent where everything they do is moronic and their character is annoying, and then the ones that fit into more realistic shows where they're the idiot only by the standards of others (and, often, these ones are more likely to display other characteristics; their "idiocy" might come from being naive, hyper, driven, and whatnot).
Personally, I've gotten too annoyed with the often-high-pitched girl common in anime who shows no interests outside of helping others, -insert male name-, and their abhorrent klutziness. They feel too useless and I don't find them funny.
