I see it as a special case of "the fallacy of everything related to high school statistics".

(Okay, so I don't really agree with the answers the post gives. But I think it's bringing up an interesting point, and hey, this is only discussion. Possibly if we had lots of high-quality math posts, I would feel differently.)

My personal feeling is that where statisticians go wrong is that they think of their problems, not as something you solve, but something you use tools on. But I'm not sure I can articulate this feeling more precisely than that.