Well, on the case of state prosecution, demanding money to avoid being reported is a crime of extortion. Companies could no longer do what some lawyer firms are currently doing. They are able to do it now because there's a loophole for lawsuits. There's no such a thing for state prosecution.
As for companies not seeing a gain in there since they have nothing to gain, that's pretty much the whole idea. Piracy, as I personally look at it, is a public crime. Pirates will pirate from all sorts of industries and all sorts of companies, big or small. Books, music, software, movies, all and from all sorts of copyright holders can exist on the same pirate hard-drive. Meanwhile, their reach is universal, once they directly or indirectly -- through P2P -- share illegal content. As an illegal activity, it has all the ingredients to be classified as a public crime. With the aggravation that generations of youngsters are being educated into a world were copyright infringement for digital content is not legal, but everyone does it and so it's sort-of-ok. In this scenario there is no place for companies to support themselves the costs of lawsuits, neither for them to collect any compensation. This is what we have governments for; It's clearly a sociological problem, a deviant behavior that is becoming too aggressive, before it can even be considered an economic problem. This is particular relevant to small copyright holders who see their lawful rights trampled on a daily basis and cannot support the costs of lawsuits.
As for the state ability to investigate, internet crime is already an area of investigation well developed. I suppose some of these tools could be used. It's a fact that states currently hold the best tools and resources, and are covered by the most amiable laws, to conduct proper investigation. What I don't realistically expect is for this solution to end piracy. But instead to put it into perspective, help reduce its numbers and provide a legal framework that truly protects small copyright holders and middle-man business such as retailers.