Perhaps another way to look at it is to say that “every past apocalyptic prediction was false” is tautologically true?

A statement can be a trivial tautology, as in “P or not-P”. Or it can be a more complicated form of tautology, such as a “self-referential” tautology, as in “This statement is true”. The anthropic principle is then just a name for the claim that there is also another complicated form of tautology alongside “self-referential”, one that we might call “indexical”.