If this were even remotely thought to be the real thing, the Vatican would have it locked away for no one to see
Locked it away for no-one to see? The cross was distributed to so many churches that Erasmus (very slanderously) said that there were so many relics that they must’ve used the entire forests of Lebanon. (In truth, they total only about 100 kg; it takes 100,000 grams to make 100 kg!)
People just tend to ignore such relics; the Sacred Chalice (Holy Grail) has been in Spain since the third century; it’s currently in the Cathedral of Valencia, where it was restored by a bunch of Frenchmen at the time when a boy named Arthur was Duke of Brittany while John I usurped the British throne.
The crown of thorns lies in Cathedral of Notre Dame; the sacred burial cloth lies in Turin; over the tomb of Jesus, a great Church is shared (unfriendlily) by Orthodox and Catholic priests.
The Vatican doesn’t have imminent domaine rights over artifacts discovered In Israel and purchased by Israeli jews. Same as the Israeli Supremes ruled that the State of Israel also didn’t have imminent domaine rights. From Samuel Clements/Mark Twain’s account, Clements found Jerusalem, and Israel in general, a barren wasteland in 1867. It wasn’t until after the Jews returned to Israel post WW2 that excavations began in earnest and the bone box was unearthed, and subsequently sold to Israeli native Oden Golan in 1976. Had the State of Israel won the court case against Golan, there would still have been a battle royale between Israel and Rome over who had rights to the artifact and Israel would likely have won.