The groove on the kingpin is not tapered. It is quite long, much longer than the pin width. The pin is round and has a flat side with maybe a very small taper to it. The pin should go in to where the threads are showing through. Then some light tapping should move it more to be able to use the threads. You don't want to hammer it hard, that would be a big mistake.
If the kingpin is off by just a few degrees, and it is tight, you can have a problem like this. The hint about the tapered drift is a very useful thing to try.
