Doors – Why can I no longer turn the knob of the self-locking door from the inside

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.

I dissembled the knob to figure out how the privacy lock mechanism works.

The slider moves a sleeve around the square axle between the door knobs.
When the round part of the sleeve is around the axle, the axle can turn and the door is unlocked.

When the square part of the sleeve is around the axle, the axle can't turn and the door is locked.

The little metal flange I found under the knob outside the door is supposed to be able to turn and work the slider.

At this point I realized that it had all been installed incorrectly. The two parts need to match up like this:

The mechanisms must have been installed in a different way such that they didn't match up correctly. When I went to put them together properly, I found out why. The gold pin is so long that it doesn't fit when installed in the proper position. When I tried to assemble it with the parts aligned properly, there was a 1/8" gap.

The lock manufacturer probably made the pin oversized so that the lock could be installed on multiple width doors. For doors less than the maximum thickness, the installer should trim the pin to the correct length. When the pin was untrimmed, it couldn't be installed such that it was in the correct position. The person that installed this knob, found a way to get it to fit, but the "unlock from outside" feature was non-functional.

I used a pair of pliers to cut 1/8" off the end of the pin.

When I re-assembled the lock and knob, the privacy feature works perfectly. The door can be unlocked from the outside just by moving the protruding metal nub under the handle with your finger.