Half of the circle pieces on this puzzle are equivalent to the centers on the Dino Octa. But the other half make up two sets of what are equivalent to Pyraminx edges.

Is this somehow related to the fact that the FTO is, essentially, two mirrored Octaminxes (truncated Pyraminxes) combined into one puzzle?

If we were to extend an FTO into a stella octangula, it would need new pieces to fill it out - I can't visualize them, but are those pieces represented by the circle pieces on the CFTO?

Comparing the Pyraminx (which I view as puzzle with two layers per axis) and the Circle FTO (which has three layers per axis) does not make sense. But you are right when you compare the Circle FTO with the Master Pyraminx. No, they are not. The single type of missing pieces is equivalent to the corners of the Master Skewb and this type of pieces is not present on the Circle FTO.

Think of it this way: the FTO is an unbandaged Octaminx. The unbandaging adds another layer. If we unbandage an Octaminx, and then re-bandage it along the previous cut lines, we get another Octaminx with a different orientation. So the puzzle is two Octaminxes in one.

So, what I want to know is this: is there a correlation between this double Octaminx nature, and the fact that the circle pieces make up two separate sets?

Anyway, you're right about the Master Skewb corners. Don't know why I couldn't see that...

Edit: Now that I think of it... it HAS to be related! When you turn one face on the CFTO, the circle does not turn with it, which means that the turn is only affecting circle pieces belonging to the other "set" (since all FTOs have 2 "sets" of sides which can never intermingle). So the Circle FTO, essentially, is the same as solving 2 Octaminxes + 1 FTO.