What you've described is a little confusing i.e. "the key that holds the hub that holds the belt pulley".
In general helical gears are pinned or keyed in place so they can be driven by the worm and turn the auger shaft arms which have the blades bolted or fastened to them in various ways (shear pin, hinch clip, bolt).
Worms are part of the driving shaft or keyed to that shaft.
Impellers can be keyed and usually double pinned.
The main driven shaft extends back through the intake housing where it's connected to something that drives the whole assembly. That's usually a coupling to a shaft and driven pulley or directly to a driven pulley. Those are also usually keyed and or have set screws.
All the components are mated to the auger drive shaft or blade drive shaft, no components turn freely on their shafts.
The pulleys need to turn with their shafts also without slipping so get the same keys, pins or set scres to lock them in place. There's no clutch mechanisms to slip the pulleys. They are engaged usually by a tensioner pulley controlled from the handle bars.
The tensioner tightens the belt and the belt then drives the pulley on the auger drive shaft.
If all the keys, pins and set screws are in place the auger drive assembly will turn as a unit without slipping anywhere. The impeller turns usually at 1200 rpm, the blades reduced to 120 rpm.
It looks like you've checked all the keys and pins in the auger assembly and probably the key for the large drive pulley in back of the housing so all fine. If the driving pulley from the motor is ok then possibly your down to the belt slipping.
If the belt was slipping but turning the impeller at 12 rpm the auger blades would turn at 1.2 which barely moving and might not be noticed. It's easy to imagine that someone could look at the impeller and on a quick look not see the blades turning as usually with the belt slipping quite a bit.
trouts