a trick of flea market carpet repair to fool the eye
one of the "tricks" of carpet repairers in trying to repair antique carpets with naturally dyed wools [when a matching color of naturally dyed antique wool is not available to them] is to take three different wools of almost the same color and untwist the very slightly different colored wools and retwist them into a new yarn which more easily can be used to repair a small defect of naturally colored wool. it turns out that even synthetically dyed wools put together that is retwisted into a yarn is less jarring and much much less apparent in making the repair.
i think that this might support the need for constant minor variations which i believe occur in natural dyed wool.
Like you, one of the things that I find attractive about antique rugs is "micro-abrash" - minor color variations that occur from one knot to the next. More recent rugs usually don't have this, and their colors are sometimes described as "flat" or "lifeless" for this reason.
I don't think "micro-abrash" results from dyes being natural rather than synthetic. In either case, the dyes are in a solution that is pretty much identical everywhere. If that's true, every bit of yarn sees the same mixture of chemicals, each chemical at the same concentration throughout the dye vat. Clumped yarn may prevent the dye solution from being well stirred at the interior of the clump, but there will rarely be variation between spots that are very close to each other along the skein.
My guess is that "micro-abrash" results from handspun wool's variable thickness. The color intensity at the center of the dyed yarn varies because the thickness of handspun yarn is not uniform along its length. When we look at a pile piece, we are looking partly at the ends of the yarn. That is, much of the surface of a pile rug consists of cross-sections of the yarn at different spots along its original length. In a segment of one color, adjacent knots were adjacent segments of the skein.
Since the use of natural dyes and of handspun wool were dramatically reduced more or less simultaneously, "micro-abrash" became unusual at about the same time that the use of synthetic dyes became common.