So all guitars with a natural resonance of G will maybe ring out and exhibit this outrageous phenomenon.
I don't know - I'm taking a stab but it doesn't seem unreasonable.
My guitar 'wants' to ring at F, and I hear your G-pitch overtones on my guitar but they sound natural; just part of the sound of the instrument.
If it's easier for your instrument to resonate at certain pitches than mine maybe there's something in that?
Like I say, though...A stab.
We've all had it where the bassist plays some note and the unattended snare drum goes mental.
I recently hung some of my guitars on the wall and I'm very aware of them ringing when I talk or listen to music there.
The reason I keep mentioning the saddle is because they are fussy as hell if not quite right.
They can/do become worn over time, and running a different string gauge from what was on it at setup can make all the difference too.
I'm thinking If I wanted annoying pronounced harmonics I'd impede the movement of the string slightly (pinched harmonic).
Seems to me nut, frets, and saddle impede string movement and the first to go would be the fundamental.
and the range of cases where you observe this rule out, or at least reduce likelihood of, it being caused by nut or individual frets.
Thinking about it, it might be worth just checking out single note sustain, with all other strings muted.
For example, fret 5 on D string (g pitch)...Just give that a pluck and see how it sustains.
Does the fundamental fade out abnormally early, leaving the first harmonic (octave above) behind?
I mean, it should do exactly that but, I guess, the question is how quickly?
I did make a guitar from scratch but I'm no luthier. I'm just throwing out things I think I'd want to try.
If anyone can tell you, or me, better, that's very welcome!