You somehow expect a private industry will do it even better? That's hilarious.
It might, in theory. For example, lets pretend that BP is allowed to purchase a warranty on their emergency shut off valves. That warranty makes the maker of the shut off valve liable for the valves failure. The company that sells that warranty is risking billions of dollars in losses if BP fails to maintain the valve. Being greedy, the company puts several rules in place to make sure BP maintains that valve, so that they don't lose several billion dollars. The company knows that BP will try to lie to them, so they require third party validation.
Yes, the company selling the warranty, or the third party doing the inspections can be lax, but so can the government. In the case of the warranty, if the inspectors or warranty company are lax, they lose everything. With the government doing the inspections, if they are lax, they lose nothing.
Now, that is all theory, and real life hates conforming to theory. And this theory would require changing some rules, such as not allowing limited liability, and not allowing the liability to be sheltered through corporate shells and such. The owners must be at risk of paying the full costs to the limit of the total value of the company. I think it could work, but I really couldn't tell you if it would.