The Accent's a Little Dodgy, Though
Speaking of origins, I find it interesting that in NCIS: Los Angeles, where the team leader and, to a lesser extent, the other field agents specialize in undercover work, yet no matter what identities they take on, they never seem to change their appearance, mannerisms, or voices. (The fact that one of them has a large birthmark on the white of her eye may also be a problem. How the impostors' faces are seen everywhere is a necessity of production so that they can be recognized by the viewers.) I can imagine that fake accents would sound silly (something that they don't mind in Burn Notice nor Leverage), but the NCIS franchise is already more comedic than Law & Order or CSI: is supposed to be.
Also, I don't have a sense of how you can tell that an accent is fake and/or silly-sounding, unless the listener is already fluent with the real version of it. (This doesn't seem like it should ruin the mood nor break the suspension of disbelief any more than characters hideously mispronouncing foreign-language words, which happens all the time on the serious shows mentioned.) For instance, my brother is annoyed by Ms. Emily Procter's fake Southern accent, because she only grew up in North Carolina. This is another area where I really can't understand how humans' brains function. Personally, I find myself cracked up by real people, such as the lodge-master in the History Channel's Freemasons say "We're not Satanists, really" special who looks and sounds like Sean Connery, or the Louie Armstrong guy from the Bermuda Triangle episode of Destination Truth. And I've seen half a dozen Daily Show correspondents who've taken advantage of location, location, location to play serious characters on Law & Order, including at least two (the ones who are married) on the least silly one.
On another related note, my degrading ability to process audio gives me even more difficulty understanding what people with foreign accents say on TV. But it still seems like it would be a bit insulting to be one of the native English-speakers on nonfiction shows who're subtitled just because they have Southern or British accents. . . .