Hi Jack,
The word for garment kuttonet is often used for the priests' clothing but it is certainly not always exclusively used of priests. It is used of Joseph's fancy outfit (Gen 37), Tamar's clothing as a princess (2 Sam 13), the clothing of a royal counselor (2 Sam 15:32) and the bride's attire in the Song of Songs (5:3). So there is nothing distinctively priestly about the word. It's a term for an extremely posh outfit. And I think that is the point: God replaces Adam and Eve's uncomfortable and impractical figleaves not with functional sackcloth but with a leather jacket and a fur coat, as it were.
I know that many fine expositors, past and present, make a big deal out of the fact that animals had to die to supply the skins, and see that as an anticipation of blood sacrifice which covers our sins; I respect their opinion, but I don't think the text particularly highlights that aspect (for example, we're not specifically told about God's killing of the animals). In my view, the focus is rather on the fact that God covered their nakedness with fine garments, not merely a bare minimum. But I'm sure there are many here who disagree on that particular point.