I'm not even sure the rare animals are anything all that special anyway. Take the Ornery Karakul. It's common leaving is Milk and the rare is Fleece. The Lost Lamb is just the reverse. Common is Fleece, rare is Milk. Now, maybe the Ornery Karakul would be special if the Milk was rare in general... But it is the common leaving from both Aurochs and Island Nanny.

There are plenty more examples of this as well - "Rare" leavings being common on the weather/time-based creatures... but these are also common on common animals. And it isn't like these special animals can drop 2+ of these items. They are still bound to drop 0 or 1.

I recently released some of my more "rare" animals in order to balance out my leavings. We can get up to 40 leavings per day from 20 animals with 9 types of leavings. So, ideally, we'll get something like ~4.5 of each type per day. In reality, this equates to balancing out your creatures so you get 5 each of 4 types and 4 each of 5 types.

What we don't know is if these special creatures drop the rare item at a higher rate. I suppose we'd need someone to have a pasture with all common animals and track what they get for a few weeks and maybe we could have that data point. I think it is a good assumption that special creatures can't have a worse drop rate. But the fact remains, there is nothing "rare" about swapping the rare leaving with the common leaving.