The study says it controlled for age, so that may mean it took into account the difference between September and August born dc and the fact that September dc are advantaged in other ways - it's not clear.
I suppose you would need to do a breakdown by month of birth, for example, and see if taller September borns do better than shorter September borns (controlling for all the other factors as well).
Interesting that this only applies in larger schools. Yet another factor to feed into school choice!