The shoulder stones were a memorial. The breastplate was something more intimate. The Targum Pseudo-Jonathan on Exodus 28:29 insists that Aaron bore the names of the sons of Israel upon his heart each time he entered the sanctuary — for a good memorial before the Lord continually.
The shift from shoulder to heart is the whole teaching. A shoulder carries weight. A heart loves. When Aaron walked between the altar and the curtain, the twelve tribes were not only lifted above him but pressed against his chest, warmed by his own pulse. The Sages taught that a priest who did not feel his people as he served them was not really serving. He was performing a job.
The word continually is pointed. The breastplate did not come off at the end of a rite. From the moment Aaron entered the sanctuary until he left, the tribes rode over his heart — a permanent reminder that the service was not private. A priest alone before God is a contradiction. He arrives with Israel, or he does not arrive at all.