- Based on a review by Kolb et al, 2012, it seems that "the brain is finished developing by 25" refers to the point when synaptic pruning in the cerebral cortex levels off, on average. However, the prefrontal cortex, the region most unique to humans and involved in executive function, develops in this way well into the third decade of life.
- The above review cites a study by Petanjek et al., 2011 that investigates synaptic pruning in the prefrontal cortex by looking at differences in dendritic spine density in tissue samples from autopsies of human subjects ranging in age from 1 week to 91 years. As seen in the figure below, significant pruning is still occurring even by age 40.
So, a critical region of the human brain is definitely not finished developing by 25, or even 35.
As for whether this process is biologically "set in stone", I think it's fair to say that is still an open question. Looking at the second figure, there is quite a bit of variance and a relatively small sample size. Moreover, this was necessarily a cross-sectional study since we can only get postmortem tissue samples in human subjects. It is entirely possible that older subjects had more pruning because they lived in a less dynamic environment. Even if we could do longitudinal studies, the question of societal trends influencing learning environment would still remain.
That said, there probably is some degree of biological determinism given that we observe similar trends in other mammals. However, as humans tend to have some of the most gene-environment interactions of all animals, and the prefrontal cortex is the most recently evolved brain region, we can only get limited information from studies in other animals.
Given the importance of the human brain development timeline to policy decisions (see paper cited in the question), I think it's fair to say these questions need more attention.
Kolb, B., Mychasiuk, R., Muhammad, A., Li, Y., Frost, D. O., & Gibb, R. (2012). Experience and the developing prefrontal cortex. Proceedings of the National Academy of Sciences, 201121251.
Petanjek, Z., Judaš, M., Šimić, G., Rašin, M. R., Uylings, H. B., Rakic, P., & Kostović, I. (2011). Extraordinary neoteny of synaptic spines in the human prefrontal cortex. Proceedings of the National Academy of Sciences, 108(32), 13281-13286.