“Ask them about integrity of the sample.”
Agreed. Belief in the Shroud as the burial shroud of Jesus is just that — belief.
But so is evolution, because it has all the same problems you mentioned.
By "integrity of the sample" I was referring to contamination of the Shroud during the many years before it was available for radiocarbon dating; and various incidents during its lifetime, such as its being boiled in oil (IIRC) by folks hundreds of years ago to try to remove the image.
Such things do not make for good forensics -- the oil may have removed some of the Jerusalem area pollens, making the evidentiary nature of the pollen's presence easier to challenge by those with an agenda.
Cheers!