“Whatever you now find weird, ugly, uncomfortable and nasty about a new medium will surely become its signature. CD distortion, the jitteriness of digital video, the crap sound of 8-bit - all of these will be cherished and emulated as soon as they can be avoided. It’s the sound of failure: so much modern art is the sound of things going out of control, of a medium pushing to its limits and breaking apart. The distorted guitar sound is the sound of something too loud for the medium supposed to carry it. The blues singer with the cracked voice is the sound of an emotional cry too powerful for the throat that releases it. The excitement of grainy film, of bleached-out black and white, is the excitement of witnessing events too momentous for the medium assigned to record them.” - Brian Eno, 1996
> for an electric guitar, a triangle wave
That is only true when the string is plucked exactly at the middle - not a regular occurrence. Usually it's more of a sawtooth wave, just without the upper harmonics.
My experience with distortion in guitars is that a huge component here is how different it can sound depending on articulation. Some pieces for example require the player to not alternate their picking, but pluck only in one direction, as the difference is audible.
This is not always the case of course as some amplifiers like those made by Mesa Boogie get their signature tone by exploiting the limited gain-bandwidth product(GBP) of amplifiers, creating an even sound that at the high end of the gain setting is largely without dynamics.
In some cases(like the BOSS DS-1) the manufacturer killed the sound by introducing a technically better amplifier chip - the original had somewhat poor GBP and poor settling time, which in combination produced a nice lowpass filter with a resonance peak at the cutoff frequency, which in turn emphasized articulation.
It's all a surprisingly huge topic.
Or simply we just like things that have some noise.