I've never been blackmailed and hopefully never will be, but I've always thought blackmail was one of the dumbest possible crimes.
Blackmail can only succeed if the victim trusts that the blackmailer will uphold their side of the bargain, but this is completely illogical: blackmail is unethical and illegal, so you have to assume that anyone who would blackmail you would also lie to you. Since there's no way to guarantee that the blackmailer won't just blackmail you again with the same information, and since they have a profit motive to do exactly that, it's only logical to assume that they will.
Jeff Bezos did the smartest and most logical thing he could do, which was to publicly call his blackmailer's bluff. That flips the equation. There's still some risk of personal embarrassment for him, but now the cost to his blackmailer is much higher and he has completely eliminated their profit motive.
It's amazing the National Enquirer has been able to get away with these tactics for so long. They definitely messed with the wrong guy.