Right now, we haven't built many private institutions that provide the same level of trustworthy information and ratings as some government agencies. I think this is because it takes time, effort, and significant resources to build a regulatory agency, and in order to be trustworthy the agency must be completely independent of the companies it regulates. A government can require its citizens to invest in such an agency through paying taxes, and thus obtain the money to pay for it relatively easily. A private organization must appeal to private individuals either to invest in the agency or to pay for it as an act of charity -- for the public good. If it appeals for investment, it must have a means of making money. If it charges the companies that it regulates, it is at risk of regulatory capture by those companies, which can threaten to withold their money if it cracks down too hard. If the agency appeals for donations, it is even more at risk of regulatory capture unless it specifically refuses donations from the industry that it regulates. The only way such an agency is trusted is if it manages to overcome these pressures and both regulates fairly and is seen to regulate fairly.
A government agency theoretically answers not to the companies it regulates but to the citizens as a whole, and should (theoretically) be immune to this sort of pressure. I often think that I'd like to move to Theory because there everything works.
So, from where I sit, the big difference isn't that government agencies are (or are likely to be) more fair and impartial than private companies that do the same thing. It's simply that funding an agency or company at a level that will let it exist and work at all is easier if you can use taxes than if you have to raise the money in other ways.
"Easier" is not the same as saying that privately run agencies are impossible. If a large enough group of people is motivated sufficiently, there is no reason that I can see that they couldn't build a private institution that independently certifies banks and financial institutions, and provides insurance for those institutions. They just have to think it's worth doing and be willing to put the time, effort, and resources into doing it.