In the Wall street Journal, Peter Manseau writes on the Shroud of Turin.
Is the Shroud real, i.e, is it the real burial cloth of Christ? We will never know for sure. Science can refute its authenticity, but never prove it.
The carbon dating in 1988 is only one of a great many scientific studies of the Shroud and, like most of them, it is controversial. In many ways the Shroud is far more of a scientific enigma than an object of faith. Surely some from a strong reformed tradition (he mentions Calvin) will enjoy schadenfreude from any evidence against the Shroud.
Manseau mentions studies of pollen in the cloth. These were done by a top forensic scientist from Interpol, Max Frei. He demonstrated that the cloth contained extinct pollens that date its presence in Anatolia and Syria to the turn of the first millennium and during Late Antiquity, respectively.
The historical record of the Shroud's existence prior to 1204 is based on documents about the Mandalion, a display of Jesus' Head which had a profound effect on Christian and particularly Eastern Christian art. This theory fits the scientific evidence well and explains the fold marks. It probably inspired the legend of Veronica's veil.
The controversy over the carbon-14 dating has to do with the representativeness of the fragments that were used for the dating. The Shroud is not an archeological specimen that was undisturbed for two thousand years. It was repaired a number of times, some documented and some not. Moreover those entrusted with its safekeeping, to the extent they believe it might be the true burial cloth of Christ, are loath to allow any destructive testing such as carbon-14 dating which requires burning a piece of the cloth itself. Thus they only permitted something from the periphery. There is evidence that actual fibers used to do the carbon dating were contaminated both by earlier repairs and by the fire. To write "these studies found, with 95% certainty, that the fabrication date of the linen of the Shroud was sometimes between 1260 and 1390," is to misunderstand the null hypothesis.
The biggest mystery (in the non theological sense of that word) is how the image was created. Not only was the concept of a photographic negative unknown in the fourteenth century, but using modern image enhancement technology scientists found that the Shroud contains three dimensional information. No known scientific process can explain how this particular image come to be on the Shroud. That is one reason why the studies keep piling up as chemists, physicists, physicians, astrophysicists and others investigate the Shroud. The vast majority of these scientists, as befits their disciplines, start from a skeptical point of view. To disprove the authenticity of a major relic would, after all, be a great coup.
From my personal point of view, the greatest benefit of all this research (as you mention) has been our deepened understanding of the physical reality of crucifixion. Crucifixion had been outlawed for a millennium at the time the Shroud reappeared in the fourteenth century. There was little accurate knowledge of the physical details of crucifixion. Those anatomical details were mostly rediscovered in the last century. One of the verses of the Anima Christi is "Passion of Christ strengthen me." Using this knowledge to meditate on the existential reality of the Passion brings home the price He paid and inspires courage to do what the right thing.