That word... I don't think it means what you think it means.
Overtones are a physical acoustical phenomenon, not a psychoacoustic one. That is to say, in order to hear them, one must have functioning hair cells in that frequency range, same as they would in order to hear fundamentals at that frequency.
Furthermore, if I recall correctly, sine waves (which I assume most folks are using for this test) don't produce harmonics.
As far as my hearing goes, I'm a mutant. I can pick up ~26kHz in my left ear and ~24kHz in the right. I'm 27. Keen hearing without hearing loss runs in my family. As far as bass frequencies go, I can hear 13Hz, but at that point, the diaphragms of my cans make a loud flapping noise that sounds like things breaking, so I was afraid to go lower.
I highly doubt it increases my appreciation of music. I may possibly be able to pick up a bit more timbral detail than most folks. But it mostly just means I have to wear earplugs to sleep. I can
hear customers walking into the coffee shop while washing dishes, so that's pretty sweet, I guess.
I often wonder how much the upper limits of hearing are physical and how much neurological. My vision is 20/480, and I didn't get glasses until I was 8. I also I have a problem with coalescing the two images in each eye into a single image (my right eye drifts inward). I have a prism in my glasses to correct this, but the first year I had glasses the prescription was reversed, so I was seeing double. Long story short, I was fairly dependent upon my hearing as a child. Then I studied music composition in college, so between ear training and score study with recordings, I developed my ear further.
Whoa, that was a rambling piece of autobiography. My apologies to anyone who read that.
My point is that, in this age of ubiquitous electronics, with lights, power cables, and a plethora of lit-up rectangles surrounding us all day, it seems reasonable that the brain would start discarding those pervasive high frequencies as unnecessary information. So perhaps instead of damage to the hair cells, most folks' brains are simply not turning the vibrations into sound?