How do I fix an exterior door knob that will unlock but not turn?
In its currently disassembled state, there is nothing locking the mechanism. You are correct that the C shaped half-circle shaft is what pulls the latch back when turned.

If you cannot push the latch back with your finger, the anti-jimmy device has had a piece break in it.

I had this happen on a bathroom door latch about two weeks ago, probably made by the same manufacturer as the shape of the lock actuator rod and latch actuator are exactly the same.

If you look at it, the lock actuator rod (the center piece you were turning) has a taper section just where it passes through the latch pull which means that during assembly, you manually push the latch back and it passes straight through.

In order to get clearance so the lock actuator rod will pass through, I pried the folded piece that they pass through till I got clearance so the shafts would pull out. I then squeezed the folded piece back together and pushed the latch out. It didn't even require taking the door off the hinges.

Replace the whole door knob/latch assembly with new and you're back in action.