I disagree. That method would get inclusions of materials of unknown provenance and ages which would then be averaged into the age of the original material. The only real sample should be a part of the Shroud that includes image... then we know that what we are getting is the "real thing."
One of the issues that perplexed the original C14 samplers was how to provide blind testing with control samples of other pieces of Linen of known provenance. They decided that they really could not do so because they had a lot of difficulty in finding other linens with the distinctive 3:1 herringbone twill.
I think they were short sighted.
The solution to this problem would be to take the Shroud samples and the control samples from other cloths and merely de-constructed them under controlled conditions and send a pile of disconnected threads for each sample. This would remove the weave tell-tale and solve the blind testing problem. Using this method, none of the labs could have determined which were Shroud threads and which were control threads. Voila! Blind testing.