Most translations of Exodus 28:12 call the shoulder-stones a memorial, and leave the word undefined. A memorial of what? The Targum Pseudo-Jonathan fills the silence. The gems are a memorial of righteousness for the sons of Israel. They do not remind God of the tribes in general. They remind Him of the tribes at their best.
That detail reshapes the entire service. When Aaron walked toward the altar, the names on his shoulders were not a neutral census. They were a case being made. Reuben was remembered through his repentance, not his error with Bilhah. Judah was remembered through his confession, not the affair with Tamar. Each tribe was lifted before God by the best thing any ancestor had ever done.
The takeaway is startling. A priest's job is not to report on his people accurately. A priest's job is to argue their righteousness, to hold up their finest hour as if it were the whole story, and to trust that God, hearing that argument, will respond in kind.