Looking at the keyseat slot in which the key slides ,it looks uniform it depth ,the ramp at the end I would think just locates the key from moving out of the hub.
It doesn't look like it would use a key that is ramped or tapered but I can see grub screw marks on the shaft which makes me think there is a screw on small hub that fits onto the thread and has a grub screw that locks to the shaft .
The thread looks too clean to not have had something attached to the thread.
This is where you really need some images of the sprocket end from another machine so you can see if a part is missing or if the part you have is original .
It's difficult to see in the image if the chain is in the centre of the cut outs for the chain.
So the only way is to see if another same model mower has a part your machine is missing or just fit some grub screws.
If it's missing a part and you aren't worried about replacing the missing part you could drill and tap the hub for 2 grub screws.
I would put 2 horizontal grub screws in the hub with the key facing upwards ,that way the shaft is balanced.
