Animals that could be sacrificed in the OT

Peter Leithart (who is on good form at the moment) observes:

Biblical sacrifices were confined to clean animals, and, more specifically, to domesticated clean animals. No wild animal ever ascended to Yahweh's altar?

What does it say about Israel that it offered only domesticated animals? What does it say that Israel detached the cult from the hunt?

I don't know the answer to that question although I have ideas. These sort of questions are going round my head a lot while reading Leviticus at the moment. Would birds count as domesticated animals though?

One passage that includes all the animals that could be sacrificed in a slightly different context is Genesis 15:8-10 when God makes a covenant with Abram:

But he said, "O Lord God, how am I to know that I shall possess it?" He said to him, "Bring me a heifer three years old, a female goat three years old, a ram three years old, a turtledove, and a young pigeon." And he brought him all these, cut them in half, and laid each half over against the other. But he did not cut the birds in half.

Of course there were non-animal offerings too (e.g. grain and wine) but I think that only cultivated crops were offered. A particularly interesting omission is fish.