Media scholars have known for a long time that the key to a popular show is to create the space for a wide variety of viewers to find enjoyment. Nearly 40 years ago, a study by Neil Vidmar & Milton Rokeach* found that conservative viewers of the satirical All in the Family were laughing along with the racist/sexist/homophobic Archie Bunker while liberals were laughing at him. The distinction is important and it may explain why Duck Dynasty is now in deep trouble.
Prior to the controversy stemming from DD's "patriarch" Phil Robertson's racist and anti-gay comments, the show could be (at least in theory) enjoyed by a wide variety of viewers. Many viewers identified with one aspect of the show or another, while other viewers learned about a part of Americana that they may have known little about. Narratives are driven by characters (the more interesting, the better) and through their words and actions, they affirm some values and challenge others. Audiences watch programs for a wide variety of reasons, and a cleverly constructed program will make room for a very diverse audience to find pleasure in watching, whether it is to identify with central characters (and laugh or cry with them), or to feel superior to them and laugh at them.
The problem for DD now is that a significant part of their audience will find it difficult to continue to watch and take pleasure from watching someone they know to be so prejudiced (even if he doesn't quite realize it himself). For those with a social conscience, Phil Robertson is no longer a harmless, sometimes even endearing hillbilly who reminds us a bit of Jed Clampett. He is a genuine redneck, complete with prejudices and a level of ignorance that is pretty much the opposite of endearing.
There is no freedom of speech issue here. It is simpler than that: It is business. While certain self-proclaimed Christians now claim to support Robertson's right to be prejudiced toward gays and lesbians and painfully ignorant of race relations, the bottom line is the bottom line, which is that A&E needs to rescue the program so that Americans who don't share Robertson's views can still find a way to enjoy the show. That won't happen while Phil Robertson remains defiantly committed to view points that many viewers find backward and extreme.
*"Archie Bunker's Bigotry: A Study In Selective Perception And Exposure." Journal of Communication 1974 24(1): 36-47.