I always struggle to understand such an anthropic principle type of reasoning.

>If any past apocalyptic prediction had been true, you would not have been around to write this sentence and we would not have been around to read it

So what? How is this [proportion of possible universes in which we can read the failed prediction] logically relevant to our universe?

It just doesn't click for me, it smells like some additional premise is smuggled, possibly the same as in the simulation hypothesis argument.

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”.