You missed the entire point of the excursion. They're not trying to steal the papers. The rock transfers "value", so things that are intrinsically valuable like gems and designer watches are physically attracted to it. But the papers themselves do not have value, they have a value that they...
Gojo was doomed to lose this from the start. There were too many loose threads with other characters needing to fight Sukuna. Kashimo's gonna use his masterpiece technique on Sukuna and lose, then Yuta's gonna try and fight him and either die or be left on the brink of death. Shoko's then gonna...