So is it really that far-fetched to think teachers could have an influence on eating & sleeping habits that could definitely affect height?
The study supposedly checks for one aspect of this – seeing if there's a cross-correlation between achievement-boost and height-boost, and thus some single general "positive influence" – and finds no correlation. But I haven't looked closely, and it's quite possible they're over-controlling to find that no correlation – which seems quite common in educational studies on ideologically-fraught topics.
I could also imagine teachers independently affecting academic achievement & height. Imagine a 2x2 model: teachers can be either 'fun/popular' or 'severe/unpopular', but also 'instructionally-effective' or 'instructionally-inept'. Fun/popular teachers might keep students in attendance, cheery, eating well at school lunches, not stressed/overworked with homework – and thus more likely to grow in height. But, only some of these 'fun' teachers (who promote mood/physical vitality) are also moving academic measures higher. Meanwhile, some of the academically-effective teachers do so via a harshness & heavy-workload that impairs height.
Likely untrue, given a few moments search. For example, "On average, teachers spend $37 a month and principals spend $59 a month for food for their students." [1].
There's lots more data on teachers feeding kids in perhaps causal amounts.
[1] http://neatoday.org/2013/09/10/report-teachers-spend-37-ever...
Given the strong correlation between some breakfast vs none, and school performance, it would not take much to affect the outcomes by feeding hungry kids little snacks.