How do these work? It looks like there a reciprocating mechanism of some kind but I don’t see any sort of chain that acts as a driveshaft.
Here’s the patent, though I’m not sure I really understand it: https://patentimages.storage.googleapis.com/5d/0a/ca/37a6df64179256/US809980.pdf
Guessing, but what else could it be - the main spring lifts the shoe/foot/sole part, this brings the wheels closer making the string loose, which then gets rolled around the hub of the back wheel (by a second spring inside). When the smooth operator presses down with their weight, the string turns the rear wheel.
However, design like this would make for a fairly limited top speed that can be achieved by the mechanism alone. Needs gearing.
This is the clearest example I could image search (gif):
… actually:
This thing also operates in the same way, just without the gravity assist:
