The styles of any crocus could be used as a saffron type spice. Those of C. korolkowii tend to be on the short side so would not provide much saffron?
The saffron crocus, C. sativus, is 'thought' to be a cultivated form of C. cartwrightianus as C. sativus has never been found in the wild by modern day botanists. In my experience C. cartwrightianus is much easier to grow, though I do give it greenhouse protection, than sativus which never flower for me.

Saffron is specific to Crocus sativus - the species has much longer/larger stigmas than do most other Crocuses, and the spice is made from these stigmas.

Beat me to it bulbaholic! :)