Throughout the past few seasons they have used Conway Twitty as a distraction from the regular plot of the episode (including some really really long interruptions).
Why has Conway Twitty appeared as a distraction in Family Guy?
Although there is no hard and fast official answer for this (at least not an obvious one), Conway Twitty used to be known as The High Priest of Country Music and held sway over his audience, often being described as 'mesmerizing'.
To this end, the makers of Family Guy have used his live performances as a distraction whenever a particularly bad taste joke or situation arises. Quoting from a wiki entry on Twitty:
In the show, one of the characters, usually Peter Griffin, needing a distraction, turns to the camera and says "Ladies and gentlemen... Mr. Conway Twitty,"
Seth Mcfarlane has long been recognized for pushing the envelope regarding how long to sustain a gag, exemplified by the inclusion of an entire Twitty song lasting well over 3 minutes in the episode The Juice is Loose.
For a more detailed breakdown of the songs and their appearances, go to the Family Guy Wiki page dedicated to Conway Twitty.
Here is an interview that Seth MacFarlane did with Kevin Pollak where he actually asks him about that specifically. http://www.kevinpollakschatshow.com/archive/?cat=202 The discussion of the overall theme of the long running joke that Family Guy does starts at around 1:03:00 The last of the discussion is at 1:11:00. The specific Conway Twitty cutaway part is discussed at 1:05:27. It's a good two hour long interview.
Basically there are two elements to the gag. One is that this primetime expensive airtime that advertisers are paying for is being used in this ridiculous way especially with (in Seth MacFarlane's opinion) such an uncharismatic performer as Conway Twitty. The other element is that a joke like this (also the hurt knee, chicken fight, Bird is the Word) is that it follows a formula of really funny-horrible-hilarious. First the audience laughs,then they don't, then they hate it, then they once again think it's fun and even funnier than at first. It has to be followed through with completely.
The intro is in the style of variety shows of 70's (Hee-Haw, Sonny and Cher, Donny and Marie, Johnny Cash). Other examples of this type of joke are given in the interview and I would like to add Any Kaufman reading Moby Dick to the list.
It's because the writers noticed a resemblance between Seth MacFarlane and Conway Twitty and decided to poke fun at him.
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