Holding the government to account is important, and getting to the bottom of matters is also important. Public enquiries are one of the most visible ways in which the government is held to account in its choice of actions.
The fundamental floor in that model is that the government chooses those who investigate the issue. That is a flaw in the approach; it needs to be redressed.
Because the government is a single organisation in itself, one of the most sensible ways to do this would be to hold choice of investigator, and outcome, publicly accountable, for example via a website which would form a dialogue between the public and the government on the way it has investigated itself.
This would be perpetuated, not set up for a single enquiry or as a "one-off".