Back on this,
I have looked again and again.
the leftovers clearly indicate that the shaft is inserted in the base, and the threads indicate that a nut was planned there to tighten it.
so i thought what if i squeez it in the vise, extend the lever a bit for more torque and give it a shot..
i inserted two bolts in the screwholes next to the center hole to give it some clearance in the middle, after that i screwed a nut on the thread that is on the leftover of the joystick shaft.
Then i used some brute force on the vice and out it came




so its out without drilling.
I know you said you did not care since you wanted to change the joystick anyway, but maybe you want to reconsider because of originality reasons.
(at least make the stick complete and just leave it in the bottom of the cabinet or something)
If you have the old shaft part for me, a new shaft with the same diameter, same length (or longer) i think i have a source that can turn it and make the new shaft like the Original.
as you can see i have the bottom part here so the length i can copy from this part and the part that you have.