Who decides whether or not to hold a public inquiry? The government of course, and they usually make their decision on the cost of a public inquiry.
But it is rather convenient when a public inquiry delves into embarrassing subjects such as :-
Never mind the fact there has been no public inquiry into political corruption after the MPs expenses scandal. Which all goes to show that we cannot trust the government to investigate themselves. Or the police: Look at how hard people have had to work at getting at the truth behind the Hillsborough disaster.
Or in other words, we cannot trust the government to determine whether public inquiries should be held, nor the scope of those inquires. Whilst the government usually does reasonable work in setting up public inquiries, and the reason for refusing to establish public inquiries is down to cost, it is not unreasonable to plan for the worst case scenario where a future government may refuse to establish an inquiry to conceal their own bad deeds.
As such the decision of what public inquiries should proceed should be in hands of a third party. An independent third party with no past or present politicians, senior policepersons, etc. Essentially a panel of the powerless.