Carotenoids and anthrocyanins make poor quality dyes, because they break down on exposure to UV light sources, and or, react with changes in pH in the environment.
Plant pigments that hold up well usually have either an amine group bond, an aromatic ring structure, or have compound double bonds in the molecular structure.
Pigments that break down on exposure to UV are said to be non-lightfast (carotenoids are in this category), or they react strongly to changes in pH (anthrocyanins are in this category) and are said to not be wash-fast.
If you look at indigo dye, for instance, it is a dimeric molecule of two indigotin monomers that come together with a strong double bond and two hydrogen bonds, and each monomer is itself made with a strong aromatic ring structure.
The dimeric complex is remarkably stable, and requires a strong alkaline solvent to break it down into the monomers again in order for it to be absorbed into the matrix of plant fibers. Once in, the very stable molecules strongly resist UV degredation (ring structure harmonics), and strongly resist changes in pH (double bond, and hydrogen bond structure between monomers, nonpolar molecule with high weight.)
The production of complex molecules like that is very uncommon, which is why quality dyestuffs commanded high prices. Many dyes of antiquity were actually metallic salt complexes that used mordants to help the dyestuffs cling to the fiber, and used strong ionic bonds to resist chemical degradation.
The ancient egyptians are often given credit for independent discovery of mordanting with metallic salts to improve natural dyestuffs, and to create synthetic dyestuffs, but china also discovered this independently.
The limited selection of quality dyestuffs is one of the reasons why "Royal purple" was "royal"-- Purple was a very difficult color to produce, because quality blue was hard to produce in antiquity. (indigo is just about the ONLY plant based blue that is both wash and lightfast.) Royal purple was produced from the extractives of a special variety of shellfish, and required special processing. It wasnt until the industrial revolution that the color Mauve was produced from coaltar, (as are all other synthetic analine dyes) that purple was commercially available in great quantities.
I can give you a short list of classical dyestuffs and even some recipies, as antique dye chemistry is a bit of a hobby of mine.