Chick-Fil-A isn't really in the meat business, they're in the restaurant business, and given that they aren't in a very tiny niche, their menus feature meat. They sell plenty of soft drinks and fried potatoes which reduce the amount of meat as a percent of the total calories.
Finally if you're going to get people excited about animal rights, it's probably better to mention cows than chickens. To me, Chick-Fil-A's ads suggest eating chicken as a lesser of two evils, without ever saying that eating chicken is evil. We also don't have a separate word for chicken meat like we do for cow or pig meat, which I think serve as euphemisms.
We do have a word with a French root for chicken, by the way: poultry. As to why the separate name didn't catch on, I can only speculate that it's because chicken are cheap to keep and so the English may have eaten them themselves.
Not all languages have separate words for meat - German perhaps being the most famous. It's slightly disconcerting to walk around a supermarket and see schweinefleisch being advertised.
I posted this article to our company #random channel and that was the first reply- better chicken than beef!
If I thought Chick-Fil-A shouldn't be supported for either their treatment of animals or for their anti-LGBTQ corporate giving, and wanted other people to stop supporting the business, I would focus on the latter in my messaging, because I suspect it would be more effective on the audience. Attempting to draw conclusions about which of their behaviors I personally think cause more suffering from this is not a good use of your time.
But any effective altruist should have their goals set higher than this, at actually reducing the insane amount of suffering that's created. It does very little good if everyone stops supporting Chik-Fil-A only to flock to another chicken company, which will be the same in almost every way, possibly minus some donations, possibly one which treats animals even worse. Effective altruism is a very different beast than simply appealing to or signaling the crowds' most popular views on ethics in order to persuade.
Do you really think Chik-Fil-A is directly causing more suffering by donating to a few charities in amounts this small than by raising in captivity and then killing billions of chickens? I see no way to reasonably argue that.