I don’t think those connections to the modern Shroud are very well-founded, but even if they were, it does not change the objection. A thousand years of silence is damning.
It wasn't a thousand years of silence. There are various reports of the acheiropoietos (not-made-by-hands) image of Christ throughout the period. The image of Edessa, the Mandylion, the Veronica are all echoes of the Shroud image in a frame work. that hid most of the body. This time period spanned centuries of iconoclasm where icons and images of all types were destroyed routinely. The image of Edessa/shroud was known to have been walled up in the City Portico of Edessa during the arab occupation starting about and only rediscovered after an earthquake in 525 - 544 or so after the city had returned to Christian hands. The image of Edessa remained there until it was brought to Constantinople in 944... where it was revealed to be a full figured image rather than just a face. At this time the inventory dropped the Mandylion from its listing and added The Burial Shroud of Our Lord to the list. In 1204 the city was sacked... and all of its relics were stolen, including the shroud.
Therefore, using your logic, Tutankhenamen is a fraud because he was forgotten for over 3200 years.
Do you really realize how few documents we have from the first millennia? Books were published in the dozens, or hundreds... not thousands. And many if not most were lost. We only have some of the apocryphal religious texts today because they were preserved in out of the way locales and survived the destructions of various book burnings. Do you know how many of the Aztec Codex survived to today... and were only talking 500 years there. Even Vatican records only go back to around the seventh century with any completeness.