Think of it, The Blood of The Lamb. Would there be any DNA material surviving? what might those tests show us? The thought of pursuing such testing is actually kind of frightening. Not that anything bad would happen, but just the thought of having the chutzpah to think one could "test" Christ's blood.
I've followed the Shroud controversy for all of my life. Personally I believe that it is, in fact, Christ's burial cloth and that the image was produced miraculaously on His resurrection. I'm fascinated by the various possible histories of this and other relics, such as the "spear of destiny," or the "Holy Rood."
One of the descriptions of the Image of Edessa, one of the possible early names given the Shroud, is as a "tetradiplon," or "cloth folded in eight." if you look at the shroud there are creases where it was long folded and, folding it along the creases results in a tetradiplon with only the face showing. This ties the Shroud to other artifacts that only showed Christ's face.
Some traditions hold that the Templars held the Shroud in secret, until their betrayel by Philip le Bel, and that their reverence for it, when folded only showing the face, is the origin of the charge that they "worshipped a head."
Gives me chills too.
I've read a couple of books on the Shroud which included the history that you outlined. I find the argument to be very plausible. Another corroborating fact is that artistic renderings of Christ were diverse until the eighth century, when the Shroud of Edessa was discovered. Following that date, Christian depictions of Christ are uniform and resemble the Shroud image.