Irina was killed, and Paul, no true detective, failed to save her. Frank, also no true detective, in fact got her killed.
Paul did not find the diamonds. They were in Caspere's bank deposit box, and later stolen from the evidence room. I guess you could say he ID'd the diamonds as stolen from a Hollywood jewelry store at the time of the '92 riots, leading to speculation as to who did the "tactical" murder and heist. And bringing in the in the mystery of the kids - who are they? The Chessani kids?
But, true, Paul's efforts in the pawn shops did finger Dixon as working against the investigation, and he did the B&E at the mansion, stealing the contracts. So yes, perhaps he has the makings of a true detective. Under my criteria, he will have to save the weak and innocent, woman or child, to become a true detective.