[edit] also see Natsu's comment for an additional thought against it being an ad for Chick-Fil-A.
As to why I think this might (I make no claims to certainty) be a PR-influenced article:
1. Chick-fil-a is mentioned in the first paragraph, no other restaraunt is mentioned until the fourth.
2. The first 3 paragraphs are a visceral imagining of what it would be like to be part of a "pay-it-forward" chain at a chick-fil-a specifically, complete with giggling cashier. Other restaurants are mentioned only in an detached and objective manner.
3. There's a direct quote in paragraph 5 from the chick-fil-a "director of hospitality", the only other quote from a restaurant owner is from a smaller local shop 3 or 4 paragraphs down. The chick-fil-a representative’s quote is a comment on the value and reasoning behind pay-it-forward, the other's is simply a factual one about a lady at his store that does it.
4. Getting a bit more abstract here, but this is exactly the kind of ideal that chick-fil-a probably wants to be associated with, "good-old down home people helping each other out small town america &c.". Note the jab at starbucks at the end?
5. Like op said, chick-fil-a took a bit of a beating in the press awhile ago. People might not remember what happened exactly, but they probably came away with a slightly worse impression of chick-fil-a. If PR was involved in this article then it's probably to try and repair that damage.
Why was Chick-fil-a mentioned first? (Puts on Sherlock Holmes cap.)
1. Maybe the writer was fishing for pageviews?
2. Maybe the writer wanted an eye-catching lead? Aggregators pick up the first sentence.
3. Is it a coincidence this is trending on a Sunday after millions of Christians just ate Chick-fil-a after church? How many priests/pastors injected this story into the sermon today? How many Christian moms did a "share by email" on this? A ton!
4. Sometimes these stories take months to write. Maybe the writer first discovered the phenomena via a Chick-fil-a related conversation.
5. A good article must have quotes. It's possible the Chick-fil-a quote came first which got the always-difficult first paragraph rolling.
6. Maybe this was an underhanded jab at Chick-fil-a patrons. "See, it's not just chicken-eating Christians that are generous." NYT's political slant is no secret.
7. Maybe the writer just likes the chicken? Is that so far-fetched? LOL.
Those are just as plausible, if not more plausible, in my opinion.
On to your points:
1. I don't really see why chick-fil-a would get more pageviews than another restaurant so I'll skip this.
2. Of course the writer wants an effective opening, but the "you're part of a pay-it-forward line" fantasy could have easily been written at a generic restaurant.
3. Other people have said it but chick-fil-a is closed on sundays. Also I don't really get what this has to do with how and why the author wrote the story.
4. I'm pretty sure a fluff piece on people paying for each other's fast food didn't take a month to write. And I think the author probably did discover the phenomenon after a chick-fil-a related conversation, one with a chick-fil-a sponsored PR representative.
5. It's impossible to say assuredly whether any article was PR influenced but that quote seemed awfully convenient.
6. You seem to be very concerned with christianity when no-one else has mentioned it, least of all the article. I would not personally assume that chick-fil-a patrons are particularly christian, it seems strange to me that this is immediately the thing you jump to. Same thing goes with the source's political leanings, I don't consider fast food to be a political matter and I don't think NYT or chick-fil-a do either.
7. Sure, but if that's the case it's interesting that most of the other restaurants mentioned are coffee shops and a bagel cafe (which aren't really chick-fil-a competitiors) rather than popeyes or mary brown's.
so in conclusion:
1. I'm right
2. You seem to weirdly have tied up a fast-food restaurant into some christian right wing identity you hold and it's preventing you from acknowledging the very really possibility of PR manipulation in an NYT fluff piece.
Anyway, you made me look it up because I wasn't sure exactly what happened. As far as I can tell, this is what started the Chick-fil-a controversy. I believe this is the full text - http://www.bpnews.net/BPnews.asp?ID=38271
In particular, near the end of the interview:
"We are very much supportive of the family -- the biblical definition of the family unit. We are a family-owned business, a family-led business, and we are married to our first wives. We give God thanks for that."
Which I suppose is why you think they need ongoing PR help in the New York Times. My first wife, we won't talk about her! LOL.
(I couldn't see a reply button on your comment just now, and now I see there is one. weird)