Well it didn't work in this case even though he was rich, as he won't get his name cleared in that way as the woman is dead and so can't be taken to trial.
Most people when accused of a crime and then not prosecuted try to forget about it and get on with their lives. To start up prosecutions against the people / persons / organisations who accused them in order to officially clear names will be very time consuming.
Remember the victim in rape cases is a witness, so you'd be looking at the ability across the board for all people accused of crimes to take any and all witnesses to court in the event that a prosecution is not proceeded with.
That seems like a waste of public money to me and would be very time consuming. Additionally there is the issues that many cases are not proceeded with due to a lack of evidence, like this one seems to be from the article, and so I mean you're opening up a very expensive very massive thing here.