I believe that that piece of wood was from a manger that Jesus slept in as an infant about as much as I believe anything that comes out of Hilary Clinton’s lips.
I doubt that Hilary would believe it either.
Your original post called into question the appropriateness of relics altogether, without doubting their authenticity. Your second post calls into question the authenticity of these particular relics—which is a different question.
I am inclined to believe in their authenticity—but that flows from reasoning based upon the authenticity of the second chapters of Matthew and Luke, which I do believe. I recognize that many do not.