Carbon dating is pretty accurate back to a few thousand years, but none of the radiometric dating methods are “exact” measurements. They all require certain assumptions and predefined reference points in order to make a date estimate.
The problem with the shroud dating is that the assumptions may have been wrong, and then the dating wouldn’t be reliable at all. They assumed that every section of the shroud was original, but some fabric seems to have been added when repairs were made. So, if they tested the repaired portion and assumed it was original, the dating wouldn’t be worth much.
That’s correct. The Shroud was severely damaged in a fire in 1532. It was repaired by a group of nuns called the Poor Clare Nuns and they used patches to repair portions of the cloth damaged by fire. I suspect when they did the first carbon dating, they dated those portions of the newer cloth. The image on the cloth, however, was miraculously untouched by the fire.