Chick-Fil-A really has no choice but to claim to abhor gay rights and marriage, because in the right wing southern Xtian states like MS, KT, AL, etc. Chick-Fil-A is considered fine dining. One has to consider the economic situation of the right wing folks that live there; middle class righties in those states just eat hot dogs or Mac-and-Cheese and go out to McDonald's for fine dining, whereas the elite upper wealthy class dines at Chick-Fil-A.
When an eating establishment is revered as a culinary creme-de-la-creme by so many right wingers, the last thing such an establishment should do is insult its patrons by supporting something those patrons consider deplorable. After all, what would happen if a right wing couple in AL, KT, MS, etc. walked into a Chick Fil-A, having saved up for so many months to afford to dine there, and saw a gay couple dining at another table. To say the least, they would pissed to the max.