The Chick-fil-a example is pretty funny. I'm sure when they started business, aligning themselves as some sort of Christian family friendly establishment wasn't really a controversial political stance. Their throw-back style and nods to Christianity (closed Sundays) were probably seen as cheap PR to ingratiate themselves to the general public.
Unfortunately for them, times have changed, and evangelical Christianity doesn't enjoy the same broad general tolerance that it used to, largely due to it's role as being a bastion against LGBT civil rights.
My reading is that Chick-fil-a is stuck between a rock and a hard place. They don't want to repudiate their evangelical, and largely anti-LGBT base, but they also don't want to be so tightly tied to the losing culture war over the issues of gay rights.
Not sure what the future holds. As LGBT rights increasingly become a settled matter of law and culture, will the evangelicals become more or less virulent over the issue?
As a matter of business, I imagine Chick-fil-a will slowly modify their "christian values" theme into a more benign version that is increasingly silent about hot-button cultural issues. They'll probably keep the things that make them seem quaint, as a matter of branding, like being closed Sundays and the "my pleasure" response, but drop the toxic assets.
