We're going round in circles a bit here and I'm not sure it's really helping.

Perhaps I'm approaching this from the wrong direction.
If on my keyboard I play a C1 I'll get a note that's mostly C1 with some harmonics due, e.g., to where the hammer hits the string. (On a piano, I'll actually get multiple very-close tones, because each string for one note is tuned very slightly differently, a kind of chorus effect.)

If I want to add more richness to the sound, I might play a C2 at the same time. Probably a little less loud, but still there. Maybe also a little C3 ... why not, it's all still the same note. To some extent it can be done on a bass also, depending on where on the string you pluck or hit. I little fifth also richens things up

There tone are all closely related to the original C1. In this case I've added them by playing more notes, but they can also be added by distortion, provided it's done gently. A little bit of low order harmonics can be added just by going through a valve that doesn't have large amounts of negative feedback, or by some cabinets, or by tape saturation. The important thing for fatness/fullness is moderation. You definitely saying you don't want harshness, but gently distortion that add just a little low-order harmonics add richness.

When you look at suggestions from Mario, Ray and Dave, some of what they're suggesting is doing just that. Adding a little, subtle, distortion. Valves, tape, transformers, cabinets and older compressors create a small amount of distortion provided they're used gently. Over-drive anything and it'll sound rough. The important thing is moderation, particularly where there is more than one note being played as then you get multiplication products that usually sound awful (to my ears ... some metal fans may well disagree, though it's worth noting that "power chords" keep that effect to more moderate levels).

Jazz relative beginner, starting at a much older age than was helpful.
AVL:MXE Linux; Windows 11
BIAB2025 Audiophile, a bunch of other software.
Kawai MP6, Ui24R, Focusrite Saffire Pro40 and Scarletts
.