We say it and sometimes mean it. Redo the carbon dating. But what lab director is so crazy at to want to do it? Suppose the results turn out about the same; it would be 1989 all over, but this time on steroids. Consider the Hungarian Pray Manuscript. In the eyes of many people it proves that a 13th or 14th century provenance is wrong. The HPM is so convincing that people, forced to abandon the patch theory if the results are the same, will latch onto the notion that the resurrection added carbon 14 to the cloth. They will do so by the millions. And there are all the questions about Thymol, Vanillin, the Manchester Mummies, etc. etc. etc. A lab director knows that even if every i is dotted and every t crossed he or she will face years of harsh public criticism and condemnation. There is no percentage in that.
And what if the results turn out to be first century? Every skeptic alive will be crawling through the details looking for a bribe or evidence of swapped samples. Some will even be trying to prove that the shroud is a first century forgery made with a heated brass statue and some leeches employed as blood dispensers.