Coaster brake hubs depend on a Bendix clutch. The core elements are the helix on the driver (where the sprocket attaches), and the clutch that rides on it.
When you pedal the helix drives the clutch to the right to engage and drive the hub shell, when you coast, the viscous friction and turning shell drive the clutch to the left, and when you back pedal, it presses into and expand the brake shoes.
So there are two common causes of failures like yours. The first is wear and low friction, allowing the clutch to slip rather than bite and positively drive the shell.
The second is sticton at the helix, causing the clutch to turn with the driver rather than stay still and be driven to the right and engaging.
Different hubs use various methods of providing drag on the clutch, so it won't turn with the driver. I can't tell from yours, but it probably relates to those two tabs on the left end of the driver. Or it might simply be wear, or too much sticky grease on the helix.
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FB
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Last edited by FBinNY; 08-26-17 at 10:25 PM.