A string fixed at both ends produces harmonic sounds because of its particular structure. In order to have a non-integer overtone the ends would have to move up and down, which by construction they can't. Similarly for wind instruments: the air stops at either end and is reflected back, and a non-integer overtone would require changing the length of the tube (or sticking holes in it to allow the pressure to go to zero at the hole, effectively creating an artificial "end" of the tube).
By contrast, a freely vibrating bar (not fixed at the ends) does not have harmonic overtones. To make the bars of a xylophone, marimba, or vibraphone sound nice, you have to cut out a little "scoop" shape from the bottom of the bar to force it to vibrate such that its overtones match up with integer multiples of the fundamental frequency of the bar.
As you say, most sounds in nature do not have a harmonic spectrum, so if a fan did I would find that surprising and interesting.