I do see where the general objection to the "current kills" comes from, even if their reasoning got a bit muddled.

All else equal, twice the voltage yields twice the current (and 4x the power).

So while yes, it's the current that kills you, in the majority of circumstances high voltage = high current. Outside of explicitly current-limited setups (e.g. a Van de Graaff generator or a desktop ionizer), most higher voltage devices will kill you more easily.

And so the "voltage doesn't kill" meme more often than not (IMO) gives laymen an undue sense of safety when working with higher voltages, at the benefit of appealing to the more electrically-inclined crowds sense of "being precisely correct".

Just look at the dozens deaths a year from people messing with microwave transformers, trying to make some pretty wood lightning art, for an example of this.