But would it really be any better if the organisation was made a statutory authority
Yes a true dilemma between the rorts of privatisation and the mediocrity of a public service institution (although there are some notable departments run very effectively and innovatively). The difference between the solid though dyed-in-the-wool Departmental Secretary vs the short term personal gain and dividend "is all and everything" intellect of new-age executives. Just look at the maginificent collapses and failures of corporations and privatised government utitilites (Energy, Telecon, Hospitals etc) that failed to manage future-driven infrastructure and the people capability needed to manage it.
Let us just hope that the same will not happen to water (purification) utilites - Pure today and Poo tomorrow. And what of safety and other institutions; consider the implications of privatising CASA, Police or Courts?
Long term stategies, low turnover rate, quality over quantity ethos will return when the toe cutters get found out. Problem is, the self-propagation principle by Board and Policial entities, magnificent survivability and unaccountability.