So do the Mormons and Jehovah's witnesses and they are both noted to be cults. Mormons teach polytheism and Jehovah's witnesses teach similarly whacked things that claim to be christian but that pervert the truth. They're viable religions, they just aren't Christian, yet they teach the resurrection of Christ. That isn't the problem. The problem is what they teach and espouse that perverts that core truth into something that bears no resemblance to the Gospel.
I believe that the Shroud is a genuine burial cloth of a Roman-style crucifixion victim. This is not a matter of my faith but of science and history; stuff that I am highly confident about. I also am able to infer from history and scriptural content that it is the burial cloth of Jesus. I cannot prove that but I am reasonably convinced that it is so.
And Mormons are reasonably convinced that something called Moron(i) appeared to Joseph Smith and prophesied in King James english. Something I don't think you'll find anywhere in the Greek; but, ok.. lol. Reasonableness isn't the rule for faith - nor are trinkets. And while you don't seem to be making that argument yourself, the Roman Church uses it as though it were some grant of authority to them for posessing it.. just as they have countless othere claimed relics. Whether it is overt or covert matters not. Science has faught with itself on this matter and still can't prove the case. If they can or cannot, it matters not a whit to Christianity. It may matter to people stuck on trinkets; but, that isn't Christianity. The treasure of all Christians is Christ himself - not discarded remnants which have no spiritual worth or significance. So, while I honor your search for the truth of the matter, I stand on my prior statements.