One of the problems is that the Ombudsman's oversight of the OIA is entirely complaints-focused. While the Ombudsman can hold practice inquiries into government agencies under the Ombudsmen's Act, allowing them to correct systematic and cultural problems which have arisen, they are powerless to make similar inquires into the Ministerial offices - Ministers having carefully made themselves immune to the Ombudsmens Act. And as the anti-transparency culture starts in those offices and then flows downhill, with agency staff getting the clear signal from above not to be transparent, then that seems to be a pretty big oversight. But there are other people who can hold inquiries, and there seems to be an obvious one for this problem: Parliament's Government and Administration Select Committee. This has responsibility for "public governance, parliamentary and legislative services, Prime Minister and Cabinet, [and] State services", so an inquiry into how the OIA is working, OIA practice in Ministerial offices, and related integrity and ethics issues is well within their purview. And they can do it on their own motion, call for evidence and summon witnesses, and make recommendations on how to amend the law to bring Ministerial secrecy under control.
Of course, Labour has a majority on the committee (though does not control its chair), and would have to vote for it. But it would be particularly shameless for a government which promised to be "the most open and transparent government ever" to vote against an inquiry into whether it is actually keeping that promise. And if they are that shameless, they should at least be made to demonstrate it, so we voters can make an informed choice about the character of those we are voting for in October.