Egad. Before we even begin to understand the NT cultural background, we need to understand that "dinner napkin" is a comparatively late meaning of "napkin." Nap- is an element meaning a piece of cloth, and -kin is a diminutive, so "napkin" means "small piece of cloth." That can mean dinner napkin, of course, although in Australia they call such things a "serviette," while "napkin" is reserved for feminine hygiene products. In the same way, "nappy" (a diaper) is an alternate form. Middle English "a napron" saw the N move to the article and become "an apron" -- a piece of cloth worn over the outside of one's clothing to protect it. So, your story doesn't even construe itself properly in English. This is not a dinner napkin we're talking about. People were not ever buried with dinner napkins. There is no such ancient Jewish tradition.
In Greek, it was a soudarion, meaning "face cloth." Think handkerchief, bandana, headband, do-rag. It was also used to bind up the jaws of a corpse in its shroud -- a use that continued until the 19th century, when morticians began to use needle and sinew to sew the dead's mouths shut for viewing. The Ghost of Jacob Marley is wearing such a cloth around his face in Dickens' "A Christmas Carol." In John's Gospel, the note that the face cloth was found rolled up (not neatly folded) by itself probably refers to its falling away from the body of Christ when his shroud crumpled into itself upon his resurrection.
Please, please, please, stop sharing this bilge.
I suppose it's unfair to the person sharing the story, who didn't know any better and just thought it was cool. But folks: we orthodox interpreters of Scripture complain endlessly about the progressives changing the meaning of the text or picking and choosing what to believe. We say, we must stick with the Bible! Bible!Bible!Bible! And then we post stuff like this, which outrages the text.
We ignore the actual meaning, which we could learn from any good Bible commentary. We not only sail right by learning about ancient cultures, we outright invent ancient Jewish traditions. All of it, based upon a misconstruing of an English word.
How can we say we want the Bible taught, in all its integrity, when we so violate that integrity by the crap we flat make up? I am ashamed to say, a lot of us just don't care what the Bible actually says, we just want our fancies given authority to overrule other people's fancies. God help us.