My understanding is that the latest DNA evidence indicates that the cloth was probably woven in India, but contains pollen, debris, etc, from all over the world (probably from visitors). This latest DNA evidence doesn't provide information as to the time of manufacture, unlike the 14 C studies. But the cloth was not woven in the Middle East, which (dare I say it?) removes some of the weight from the pro-authenticity pan of the balance. Although to be fair there is the small possibility that the linen could have been used to wrap Christ after it was woven in and imported from India.
Your statement is very much true: to negate the definitive 14 C dating studies one must propose that the dated fragment was a Middle-Age patch (even though this would have to be invisible even under a microscope, maintained the weave lines without interruption, and was not detected by any experts, including those assembled by the Church expressly to be certain that it was not a patch), and/or that after all decontamination employed by the scientists, this strip of cloth still had more than 50% contamination by weight of modern materials.
Or one can propose that the 14C dating was a huge conspiracy entered into by the Church and by all 3 scientific groups, the later of whom did the analysis independently and double-blind together with controls. Although the weave of the shroud linen was fairly unique, one of the labs reduced sample and controls in indistinguishable ash before passing it on to the people doing the dating to be extra certain of their double blinding protocol.
All explained multiple times to Jabba without any change in the discussion (expect fringe reset very shortly). But fyi.