Wooden artefacts - such as the Schöningen spears - can survive at least 300,000 years in the right conditions.
The cross on which Christ died was a real object. There’s no particular reason why pieces of it can’t survive a mere 2000 years.
No doubt, wood can survive.
But if no one really saw Jesus after the resurrection for a few days, who would have thought to go get the pieces.
While his execution was a big deal to us, was it really big enough for people to chop up the cross for relics?
I am not trying to sound like I am making fun of it, but the common sense aspects of it befuddle me.