Let me explain.
Let's start at the overlooked clue, the letters. The letters are sent out to those that are in the secret auction. No one knows how is coming other than the Auctioneer himself. He became the prime suspect immediately.
Now I'll explain what happened.
The Mechanic dozed off, after all the first thing we see of the Mechanic is lazy. He accidentally crashed his face into the controls and cause the blackout, hence why he has the bandage on his face.
The Auctioneer pushed the business man of the ship, hence why the silk hat is in the ocean, and make him seem like the suspect. The Police must have took a shot in the dark and shot the politician by accident. The Auctioneer then takes the diamond and hides it.
Now why isn't it the others? Waitress was holding something, she wouldn't have the time. Actress is rich, she has no motive. As for why she took whiskey on rocks, personal preference. Magician is too close to the diamond, if she took it she would be the first person to be check. Mafia wouldn't have the time to go to the diamond and seat again. Doctor wouldn't be treating the politician if she took the diamond.
Now here comes psychology. A diamond of such value, why would you sell it? Humans would rather keep it a secret with greed than sell it to others. Oddly suspicious. The whole auction is a fake, it was a set up to make the Auctioneer seem like he doesn't have the diamond, hence no thief would steal the diamond from him. Clever plan.