If I recall correctly, I think I read that the error wasn’t in the carbon dating but in the sample of cloth which was used for the carbon dating - it was of a repaired area and I think there were indications of woven repairs made over the centuries and the sample was not from the main original cloth where the corporal stains were located. Of course that raises more questions such as in whose interest is it to sample such a repaired area of the cloth?
Wasn’t there a later carbon dating that went back to the 1st century?
I understand that the spot sampled was repaired after a fire sometime in the 15th-16th Century (may be wrong on the date). So if the material used was contemporary to that time, the carbon dating would be from that time.
From what I gather, some of the more compelling evidence consists of pollen particles vacuumed from there. I understand that some of them come from plants that are native only to that area (Jerusalem and environs)
As to why somebody would do this, I can't speak to somebody's motives with any kind of authority. So I won't.