My ideas are also built on faith and experience. We don't have the Ark of the Covenant, either, and for a very good reason. We can't even keep a Temple, we just get the basement. We'd end up worshipping the relic instead of God. I don't believe God likes relics because of that very temptation.
But the Shroud interests me enough to follow the story.
I agree to an extent, but God (I believe) provides us with many evidences of his existence and “aides” to our faith but nothing so concrete as to make are belief irrefutable lest it is no longer be faith but fact.