I think the ancients did produce purple dye via admixture.
Various challenges with that approach (and purple dyes in general) likely affected its historical ubiquity or our perception of that ubiquity.
Regarding your specific sub-questions:
Though challenging, ancient humans have mixed dye components to create purple dye, however, more complex factors probably limited the ubiquity of this approach and, likely, our current perception of the ubiquity of that technique.
The physical and chemical mixing and application of dye components have challenges unique to the principles of mixing and applying acrylic paint, notably in the mass-produced textile context you discuss.
My basic art training hints that creating a color-fast, long lasting dye of a singular hue and applying in bulk to textiles is more challenging than applying opaque pigments mixed as a media and applied to a surface. Color combinations often muddy or dilute themselves, which is why a distinct natural source like chemical (animal, vegetable, mineral) derivations are superior as a pigment and probably more so in a dye context.
There are historical precedents for using mixtures to create purple dye (in Western culture and beyond) but it is likely that certain chemical or environmental factors limit our perception of the ubiquity of purple dyed textiles (admixture or otherwise) throughout antiquity.
Long-term instability of dye and pigment constituents for a purple hues can affect their discoverability. Defining the optical aspects of colors described in spoken or written history and matching to modern colors becomes more challenging with fewer corroborating artifacts. From Wikipedia:
A popular new dye which arrived in Europe from the New World during the Renaissance was made from the wood of the logwood tree (Haematoxylum campechianum), which grew in Spanish Mexico. Depending on the different minerals added to the dye, it produced a blue, red, black or, with the addition of alum, a purple color, It made a good color, but, like earlier dyes, it did not resist sunlight or washing.
Though the time period for the above excerpt might not be considered ancient, the techniques used could be applied further back in time.
Social constructs including popularity and marketability are also likely discriminating factors. Iconic desirability can actually hamper something becoming popular or ubiquitous when other forces influence availability.
In the middle ages, color-segregated guilds were forbidden to dye colors by other guilds. That sort of rigidity certainly wouldn't invite more inventive experiments - admixtures included - during that time period.
Though an edge case example, one possible reason ancient tekhelet (one of it's suggested color ranges being purple) dye was not replaced with an alternative replacement was the precious nature of the source that reinforces the sacred ritualistic nature of its use. From Wikipedia:
Yet, although this dye was much cheaper to obtain, the rabbis cursed those who substituted techelet with some low-priced equivalent and in fact preferred to annul the obligation altogether rather than to compromise its value
Perhaps it's possible that an absence in the historical record of a societal mass-market of a cheap or plentiful purple in ancient times is also related to both this and other intentional or unintentional limitations in the traditions and trade of dye production and human-made artifacts.
The intersection of ritual, including apprenticed traditions and closely held trade knowledge, combined with rigid societal structures (rare-colored goods as a symbol of power) and market forces further restricted by comparatively less free time or interest in novelty when compared to modern times could have simply choked out more egalitarian access to this royally symbolic hue by any number of subversive inventors or entrepreneurs.
I think the ingenuity required to source a stable purple hue - let alone devise a process through mixing - and establish a mass market definitely has had numerous challenges through the ages.