The shroud does have blood on it. And being Jewish, and dead things or things associated with dead things being unclean, I pretty much doubt they would have had the shroud nearby. Also recall that the image is not easily seen by the naked eye. It wasn’t until that guy took a photo of it and saw the negative that the image was really seen for what it was.
Obviously it would have been a relic even without seeing an image, which is why someone grabbed it (along with the head cloth - I forget what that is called, but it is at some other church, and has similar pattern of blood stains, etc.)
However, regarding the Jew’s aversion to things associated with the dead - I wonder who grabbed the clothes?
Good points, but I think, clearly even the Jews buried their dead, so the problem of handling blood on a dead person was resolved somehow. In fact, we know how it was resolved: the one who washed, embalmed and dressed the body went through a set ritual of purification. The same women, -- we know who they were, -- given the significance of the empty tomb and the preserved burial cloth -- would save the cloth.
But the aversion to blood might explain how it disappeared from view for centuries. The disciples knew it was sacred, but held back from handling or showing it without a direct need.