JKR,

Two questions that might help you diagnose:

Is there a faint firing pin imprint on the primer? 

Does the hammer fall only to the half-cock notch?

Those answers will help you determine whether the half-cock is catching the hammer.

The fact that you haven't had misfires with the pistol primers might indicate that you have insufficient striker force.  I once had a rimfire Winder musket that had the firing pin shortened to just under 0.040" protrusion.  A new coil spring didn't solve the problem, but a new firing pin with 0.055" protrusion solved the problem.

The Needier firing pin may just need a cleanup to get out anything that may have become trapped, or just a very light coat of oil to minimize friction.

CADyk

I found years ago that the inside of a Winchester single shot can trap a good deal of crud. Old lube, bits of dirt and even other stuff. My first rifle came to me due to the owner having similar trouble as you are having, but not with the Neidner pin no a fly because it is a plain trigger gun. I learned how to take it down to do a 'new gun cleaning' and was surprised to see how much was in the block- no doubt slowing the pin to prevent firing the primer. I wonder if that had ever been mucked out since the rifle was new before the turn of the century. Once cleaned and lube it soon became my favorite rifle. 
One very good feature of the original rifles over those with the so-called Neidner variety is that the pin can be removed without dismantling the whole damn action and emergency cleaning can be done in the field with nothing beyond a screwdriver and a couple of toothpicks. and oil.