I wonder if we've got folk etymology going on here. Given Dr. Bill's observation about the Latin word saeta, seta(the ae in Classical Latin words often became pared down to e in Medieval Latin) meaning the stiff bristly parts of a plant, it may have referred to the crown of thorns Jesus wore on the way to the Crucifixion and then became symbolized by arrows due to association with the Spanish word saeta derived from Latin sagitta.

I mo check out the text in the Vulgate.

Feeding s(a)eta* into the search field on the on-line Vulgate yields nothing.

http://www.lib.uchicago.edu/efts/ARTFL/public/bibles/vulgate.search.html