Many so called relics were searched for and preserved by St Helena, mother of Constantine. More were dug up by the Knights Templar over the centuries. The authenticity of any of these is highly debated
Those things don’t even matter. There isn’t a spot in the Bible telling Christians to seek out artifacts or to hold them to any esteem at all.
And rightly so. The upright portion of the cross would likely have remained in place and the crosspiece used over again. The logistics that would have someone taking and hiding the whole thing, then Helena having found it intact despite being buried for almost three centuries, is very dubious indeed. Or it could well have been that local Christians buried a cross for her to find, thus getting in the good graces of the emperor’s mother, which is never a bad place to be
I think the evidence for the authenticity of the Shroud of Turin and the Sudarium of Orvieto is powerful, and boosted, if they are (and they are) first century textiles, because possession of cloth stained by the blood of the dead was ritually unclean, and keeping those would not have been done unless there was a very good reason for doing so. Not to belabor the obvious, but a few yards of cloth are a lot easier to keep and hide than a huge cross