Yes, the fees are paid to the publishers— who themselves do not pay the authors, reviewers, and even (sometimes) the editors of the journals. As part of the NYC non-profit high society Jstor is far from the most efficient non profit— but their fault is not their inefficient spending and high salaries. Their fault is facilitating a system which is amoral and harmful to society.

Absent Jstor the universities and libraries would have an easier time reforming the system because writing a single big check per quarter wouldn't remove most of their problems (while leaving everyone outside of those institutions without access).