It's special. Just because people overload it with spurious claims doesn't alter its mathematical elegance. And seeing as how it falls out naturally from geometric figures like pentagons and pentagrams, people are likely to stay interested in it for as long as humans have 5 fingers.

Nor does it make sense for a monitor. I can't remember ever wanting to arrange a bunch of windows as a square, a smaller square, a smaller square, etc.

You must not use a tiling window manager.

I must not want my wm to give me a geometric series of ever shrinking squares.

What else does it default to, if you open a new window?

Does/did make sense. European paper size (A*) are based on the golden section (so A0 Length == A1 Width and 1/2 surface A0 == surface A1 and so on).

Not quite. ISO 216 paper sizes are designed so that if you split them in half you get the same ratio (1:sqrt(2), or 1:1.414). The golden section is so that if you take a square out from one side, you get the same ratio (1:1.618).