Thinking more, I am not sure my idea above will work. Children don't control parent positions. So to have a rotating disk controlling the feet, you need to start with that at the top. Hmm. Maybe have a rotating disk at the top, then a child of that is the puppet attached at the feet to the spinning disk (so the feet go around in a circle. Then a pin on the rest of the body so that does not move.... My brain hurts!
But note, its not that hard to do cycle layers in Illustrator moving the artwork around. There is a "puppet tool" now that allows you to distort in the same way CH does, so you can clone say 5 copies of the legs and use that tool to adjust each layer individually.

You can put pins around the place then drag them to distort the legs etc.

It might need some fiddling to get right, but the puppet tool seems pretty useful.
Ok, want some cringe factor! ;-)

The puppet structure. The wheel is a top level layer. The legs are attached using Hinge so the feet stay flat (pretty flat anyway). The body is also independent attached with hinge so the body does not fly around in cycles with the wheel. The wheel has a Transform behavior attached which is used to rotate the wheel.

The problem is the top of the legs cannot be attached to the torso. Pins etc are all ignored as far as I can tell. So the top of the legs move around in the same circle as the feet - they don't stay connected to the body.
BUT (here is the cringe part), you can put a dragger on the top of the legs and record drags while the wheel is moving to manually keep the top of the legs near the body. You just have to line up the two recordings carefully.

So there is a blend to turn the wheel around smoothly from 0 to 360 degrees, then two manual draggers are recorded separately.

I don't think this is a good solution, but it was the best I can come up with. CH only seems to allow one point to have a parent of layers, so you cannot have two separate parts of the puppet automatically get dragged in different directions that I can see. E.g. if feet are attached to parent, you cannot have a magnet somewhere else trying to attach a different part of the puppet.
(And yes, I should have put sticks on the legs to keep them straighter.)