I have never thought it was the shroud of Christ, due to the fact that it would take it out of the realm of faith and into the realm of fact.
I have some relics of saints in my possession (the Curé d’Ars & Elizabeth Ann Seton among them), and I feel like their keeper. I do meditate on them, and pray, but they are different than relics of Christ himself. His existence as part of the Holy Trinity is a matter solely of faith.
We are to rely on faith, not things. As a Catholic, I rely on my faith.
That's fine. But Jesus himself did not require just faith. When Thomas doubted, Jesus allowed him to touch the wounds and even required him to put his hand into His side:
24But Thomas, one of the twelve, called Didymus, was not with them when Jesus came.25So the other disciples were saying to him, “We have seen the Lord!” But he said to them, “Unless I see in His hands the imprint of the nails, and put my finger into the place of the nails, and put my hand into His side, I will not believe.”
26After eight days His disciples were again inside, and Thomas with them. Jesus came, the doors having been shut, and stood in their midst and said, “Peace be with you.”
27Then He said to Thomas, “Reach here with your finger, and see My hands; and reach here your hand and put it into My side; and do not be unbelieving, but believing.”
28Thomas answered and said to Him, “My Lord and my God!”
29Jesus said to him, “Because you have seen Me, have you believed? Blessed are they who did not see, and yet believed.”
—John 20:24-29
Thomas called Didymus, had to be shown proof, yet Jesus additionally blessed those who had not seen and believed, but he did not condemn Thomas for his lack of faith, but who had to be shown proof. Jesus instead provided proof for Thomas. Thomas, the doubter, was still blessed as an Apostle and a believer. Who is to say Jesus would not provide proof for later day doubters?