"Purple" as a highly valued color is in many ways an oversimplification of the chemistry behind dying, as well as the visual and secondary characteristics that would make a color desirable and/or valuable. Prior to the invention of synthetic dyes, dying required often proprietary and guild-protected knowledge of mordants, fixatives that "set" the color, as well as substances that created the desired color. Here's the real key: well-crafted dyes were more intense, held their color in sunlight and with age and in contact with the human body, and could be washed. Dye also has to look "rich" and intense for it to have value in either the historic or modern economy. The textile industry is even now plagued by the temperamental characteristics of some colors, indigo is still notoriously laborious and subject to both fading and crocking/bleeding. It's said that the Tyrian Purple became more intense and rich with age, rather than fading, which would be quite an accomplishment for even a modern synthetic process to achieve.

Natural dyes don't use pigments that can easily be combined with one another, rather you would have to overdye, dye with one color and then add another. Simply put, this is not as desirable of an outcome than working with a direct dye of the desired color. You'll get a purple or violet tone, but it will look less intense and be less attractive to the eye. This is true in a modern, synthetic dye setting as well, overdying a fabric rarely produces stable and desirable results.

I know that knockoffs of the Tyrian purple existed, and were subject to prosecution from my textile history and dyeing classes, and that Pliny the Elder went into deep detail as to both the color and process behind the dye.

I also know that violet also occupies a fairly unique position on the visual spectrum and that combining red and blue does not a true violet make (this is a gross oversimplification of color theory by a soft-scientist, apologies). I was taught that the value placed on the intensity of violet in antiquity might have been impacted by our ability to see violet light independently, although I cannot find any direct sources to support that at the moment in my textbooks.

Dying remains both an art and science to this day, the apparel industry employs a wide variety of color management and analytical tools to achieve something close to consistency in manufacturing, but it is far more difficult than one might expect from wandering down the craft and dying aisle at Michael's. Hopefully that will lend a bit of insight as to why such a rich shade that lasted was such a valuable commodity - it was as much its intensity and longevity that created the unique value.