You realise that these would include the likes of the Iraq Inquiry (Chilcot report), Shipman, Leveson Inquiry, Hillsborough, E.Coli, Aberfan...
Ok, so a few of those are hugely different - it could be argued that the Iraq Inquiry isn't serving the public, but...
Let's take a brief look at Aberfan - a coal mining disaster whereby something like 140 lives were lost (including children). The National Coal Board attempted to cover up/deny their responsibility as (I understand) they had done following previous incidents. The difference this time is that the Public Inquiry which followed bucked the trend - instead of finding in favour of the NCB it delivered a verdict confirming their culpability. Now, I can't say that previous inquiries had been wrong, but public opinion and pressure was against the "big boys" of the NCB and they were vindicated.
Now, let's look at something a little more current - the Hillsborough Inquiry. It's also a closer comparison to your scenario, too. The police - surely those that should be the most trusted of society - have been found to have lied and covered up.
The media supported their arguments and position.
Only decades of public pressure have forced the inquiry. The inquiry has found many holes in the accepted story and even those who were in the police and there on the day have spoken out against their former taskmasters. The families of the victims - who were initially recipients of the wider blame - are finally receiving some kind of justice. They've had to fight the police and the media for their justice and they are being vindicated in their actions.
The Leveson Inquiry isn't so much in response to public pressure, but it certainly serves the public interest - celebrities are members of the public too, but they weren't the only people affected by unethical media reporting methods.
The people who want these inquiries and inquests aren't always powerful and rich. You could potentially crowdfund, sure, but I expect making groups/people pay for their own enquiries, crowdfunded or not, there would become a kind of "class tier" system - minority groups and the poor simply wouldn't be able to fight their corner and uncover the truth.
Following on from that, the Government is there to serve the people and to use it's funds to do so. Therefore it is perfectly acceptable to use public funds for these inquiries, as that's what they are for!
A six month limit? Be it public or private money, six months is not enough to conquer the complexity of the issues that require such scrutiny. Either people won't bother, or many will go unfulfilled...