Because I don't like that we're all working apart from one another. If a team decided to destroy it, it could catch King's attention when one of his people dies.
I don't want to be eaten by the ground again, or killed by King. Doing nothing will make that a sure fate, but... if someone else does something that leads to King moving against us, I want to at least know it's coming.
And, it's not a good thing that we can't share information. If Kid hadn't raised such a fuss, we wouldn't even know someone found his keystone. We might have gone looking, wasted our time if we thought we saw clues to his old churches. Temples. Whatever.