Remember that the definite genitive corresponds to THE x of THE y in English. Thus, you can’t say dalta na scoile if you mean A pupil of the school, ONE pupil of the school, because dalta na scoile means THE pupil of the school – as though there was only one pupil at the school. (When you say fear an phoist for “the postman”, you actually do suggest that there is only one postman serving the place, or that whoever does the job is THE postman.)
One of the pupils is, of course, dalta de chuid na scoile, i.e. one of the “share” of the school. Like, there was a global pool of pupils, and this particular school had its particular share of them.