After removing the stigma from a flower, specifically the Crocus flower, does it eventually grow back or does the plant die?

Neither. (From personal experience growing a few saffron crocus in my garden.) Like other bulbs (or in the case of crocus, corms, to be technical), the saffron crocus has an annual life cycle. The bulb sends up leaves & flowers according to seasonal* cues.

The flowers will die whether or not you pick the stigmas. Indeed, the stigmas are usually the largest and most pickable just at the point when the flower is about to start senescing anyway. At that point the plant is about to go into winter dormancy anyway.

The plant does not die, but stays in a dormant state over the winter, then grows leaves in the spring. My experience, though, is that they will eventually deterioriate over several years unless the corms are lifted and divided.

The problem with mass production is not growing the plant: it grows readily in many areas of the world - but the labor-intensive harvesting needed. The stigmas need to be plucked individually, and carefully. Growing in a greenhouse would be possible, but uneconomical, since you need lots of plants to provide commercial quantities. (70,000 plants for 1 lb/0.45 kg of saffron, per Wikipedia: https://en.wikipedia.org/wiki/Saffron ) It's unlikely that you could speed up the annual growth/flowering cycle enough to matter.

*Saffron crocus (C. sativus) is a bit unusual in that it flowers in the fall rather than spring.