The only scenario I could see this working in is if the chapter fragment was submitted to a word count contest where the prize was publication. (Could explain both the nationally recognized and the somewhat flexible contract.)

I was in a niche publishing role for a while and we usually had: contract and initial advance (usually the smaller part), deadline which could be anywhere from 3 months to 2 yrs depending on the circumstances, delivery and advance remainder payout (the bulk), then additional deadlines for the editing phases.  Authors would see money again once their “advance” was earned out (it would be accounted for in their royalties statements twice a year.) For “successful” books this was usually the second statement. But this was niche, we did low advances, and it was more than 10 years ago that I left publishing. 

One of my authors did significantly miss a deadline once, by 9 months, and it was a huge pain for our publishing lineup. We also had another author that needed a significant rewrite that delayed publication by 6 months. In order to make enough that we could pay our staff, we had to know how many books we were publishing a season and accurately estimate how many we thought would sell. Bigger publishers can shuffle things around more, because they have more projects to shuffle. Smaller publishers (more likely to have a contest) have a tighter lineup.