But centuries upon centuries would have elapsed. To my way of thinking, the fundamental question would be whether the Holy Family’s taking all or a piece of the wooden manger would have been consistent with the Jewish mindset prevalent at the time. Breaking off a piece, or hauling the whole feed trough back to Nazareth? Clearly the Holy Family wouldn’t have just broken off a piece. Wouldn’t Joseph, as a carpenter, wanted to make an actual cradle for the Child, as opposed to using an animal feed trough? 21st Century Americans are all about souvenirs; but 1st Century Jews? So 30 years pass before Jesus becoming a public figure - and even then, his closest Apostles still struggled with the idea of his divinity. What would the trigger have been to head to Bethlehem and find his manger/cradle? And after 30+ years, who’s to say the farmer still had it or knew the right one, or whether, after 3 decades of agricultural use, it was still in existence?
The validity of the tomb at the Church of the Holy Sepulcre has been upheld by even irreligious archaeologists in terms of that site having been outside the walls of Jerusalem at the time of the Crucifixion (vs. claims for the so-called “Garden Tomb.”) But a piece of wood from the manger? Just too many questions and relic abuses from the period of that pilgrimage to Rome.