Rav Hirsch strikes again!
On the second verse you quoted (Ex. 28:38), he writes, as part of a much larger discussion on the ציץ:
...The כהן גדול (High Priest [who wears the ציץ, Showplate]) represents
the whole Service of the Sanctuary, and the inscription קדש לה' (holy
to God) on his forehead proclaims the Name of God, which expresses the
whole essence of the one God in His absolutely free personal being.
The Sanctuary is consecrated to Him, and all the sacred things of the
Sanctuary are directed to Him.
Hence the ציץ is designed to nullify
any defect or impurity these sacred things might suffer in respect to
their unswerving direction the one God. The positive proclamation קדש
לה' -- which radiates from the forehead of the כהן גדול throughout the
Sanctuary and to all its sacred things, inscribing them with the
inscription קדש לה' -- can nullify serious errors....
This, I think, about sums up his basic point -- the ציץ is a declaration that everything in the Temple is consecrated to God, and through that it can atone for errors that were made with holy things קדשים in the Temple.
I would recommend reading his whole discussion inside, because there is a lot more elaboration on this topic (and exactly how and for what the ציץ can atone), but this is really the main point of that piece.